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Linear Programming - Class 12 Commerce Maths - Extra Questions

Solve the following equation
2(2x+3)106(x2)



Check, whether the half plane 3x+6y0 contains (1,1), if so shade the plane containing (1,1).



Shade the following half plane: 5x+10y200.



Check, whether the half plane x+y+4>0 contains the origin. Identify the plane and shade it. 



The solution set of constraints of Linear Programming problem is called



What is meant by "Open Convex Region"?



Define: 'Objective function'.



Define feasible region in LPP.



Determine graphically the minimum value of the objective function
Z=50x+20y...(1)
subject to the constraints:
2xy5...(2)
3x+y3...(3)
2x3y12...(4)
x0,y0...(5).



Minimize Z=7x+y subject to
5x+y5,x+y3,x0,y0.



Dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A, while each packet of the same quality of food Q contains 3 units of calcium, 20 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and almost 300 units of cholesterol. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A?



A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws A while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws A at a profit of 70 paise and screws B at a profit of Rs.1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the given LLP and solve it graphically and find the maximum profit.



Minimise Z = 3x + 2y
subject to the constraints:
x + y \geq 8 ... (1)
3x + 5y \leq 15 ... (2)
x \geq 0, y \geq 0 ... (3).



Solve the following inequalities by graphical method:
2x+y\ge 6,3x+4y\le 12



Solve the following linear programming problem graphically:
Maximise Z = 4x + y ... (1)
subject to the constraints: x + y \leq 50 ... (2)
3x + y\leq 90 ... (3)
x\geq 0, y\geq 0 ... (4).



Find graphical solution for the following system of linear inequations:
3x + 2y\leq 180; x + 2y \leq 120, x \geq 0, y\geq 0
Hence find co-ordinates of corner points of the common region.



(Diet problem) A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30\ g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimise the amount of vitamin A in the diet? What is the minimum amount of vitamin A?



Solve the following Linear Programming Problems graphically:
Maximise Z = 3x + 2y
subject to x + 2y \leq 10, 3x + y \leq 15, x, y \geq 0.



Use graphical method
Maximize: z=2x+3y
Subject to:
x+2y\le 40
6x+5y\le 150
x\ge 0, y\ge 0



If possible, maximize z = x + 4y, subject to 3x + 6y \leq 6, 4x + 8y \geq 16 and x \geq 0, y \geq 0.



A paper mill produces rolls of paper used in making cash registers.Each roll of paper is 100m in length and can be used in widths of 3,4,6 and 10cm.The company's production process results in rolls that are 24cm in width.Thus, the company must cut its 24cm roll to the desired widths.It has six basic cutting alternatives as follows:
Cutting
alternatives
Width of Rolls(cm)
3
Width of Rolls(cm)
4
Width of Rolls(cm)
6
Width of Rolls(cm)
10
Waste(cm)
143---
2-32--
311111
4--212
5-41-2
6321-1
The minimum demand for the four rolls is as follows:
Roll width(cm)Demand
22,000
43,600
61,600
10  500
The paper mill wishes to minimize the waste resulting from trimming to size.Formulate the L.P model.  



(Diet problem): A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food I and Rs. 70 per kg to purchase Food II. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.



Solve the following linear programming problem using graphical method
Minimize: Z=60x+30y
Subject to:
2x+3y\ge 120
2x+y\ge 80
x\ge 0, y\ge 0



A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and hours for assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?



Find the maximum value of 4x+7y with the conditions 3x+8y\leq 24, y\leq2, x\geq 0 and y\geq0.



A library has to  accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weight 1 kg and 1\dfrac{1}{2} kg each respectively. The shelf is 96 cm long and atmost can support a weight of 21 kg. How  should the shelf be filled with the books of two types in order to include greatest number of books. Make it a linear programming problem (LPP) .



The taxi fare in a city is as follows: For the first kilometre, the fare is Rs 8 and for the subsequence distance it is Rs 5 per Km. Taking the distance covered as x Km and total fare as Rs y. write linear equation for this information, and draw its graph.



Formulate this problem an LLP and solve it graphically.
A Diet for a sick person  must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs 5 ans Rs 4 per unit respectively. one unit of food A contains 200 units of vitamins, 1 units of minerals and 40 units of calories, while one unit of the food B contains 100 unit of vitamins, 2 unit of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost, but it must satisfy the requirements of the sick person.. form the question as LLP and solve it graphically.



Solve each of the following linear programming problem by graphical method.
Maximize Z=18x+10y
Subject to 4x+y\le 20
                 2x+3y\le 30
                   x,y\ge 0



Solve the following systems of inequalities graphically: x+y\leq 9, y > x, x\geq  1



Solve 3x+2y\leq 12,  x+2y\leq 16,  x+y\leq 10,  x\geq 0, y\geq 0.



Suppose f is the collection of all ordered pairs of real numbers and x=6 is the first element of some ordered pair in f. Suppose the vertical line through x=6 intersects the graph of f twice. Is f a function? Why or why not?



A housewife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg of food is given below : 
Vitamin AVitamin BVitamin C
Food X123
Food Y221
One kg of food X costs Rs. 6 and one kg of food Y costs Rs. 10. Formulate the above problem as a linear programming problem and find the least cost of the mixture which will produce the diet graphically. What value will you like to attach with this problem ?



Draw the line passing through(2, 3) and (3,2). Find the sum of coordinates of the points at which this line meets the x-axis and y-axis.



Solve 4x + 3y > 12 graphically



Solve the following inequations \displaystyle \frac{2x+4}{x-1}\geq 5



Solve & graph the solution set of -2 < 2x - 4 and \displaystyle -2x+5\geq 13,x\: \epsilon \: R



A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time.

If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the maximum profit of the factory when it works at full capacity



Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs.\ 60/kg and Food Q costs Rs.\ 80/kg. Food P contains 3\ units/kg of vitamin A and 5\ units/kg of vitamin B while food Q contains 4\ units/kg of vitamin A and 2\ units/kg of vitamin B. Determine the minimum cost of the mixture.



Solve the following system of linear inequality graphically
\displaystyle 2x+y\geq 6                     ......(1)
\displaystyle x-y\leq 2                        ......(2)



A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods {F}_{1} and {F}_{2} are available. Food {F}_{1} costs Rs.4 per unit food and {F}_{2}s costs Rs.6 per unit. One unit of food {F}_{1} contains 3 units of vitamin A and 4 units of minerals. One unit of food {F}_{2} contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the mineral nutritional requirements?



A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs.250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs.200 per bag contains 1.5 units of nutritional elements A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag> What is minimum cost of the mixture per bag?



If S is the solution region of the inequalities   \begin{cases} x+3y\le 18 \\ 2x-3y\le 9 \end{cases} and (a,b) \in S, then the minimum possible value for b is



Solve the following systems of inequalities graphically: x+y\leq 9, y>x, x\geq 1.



Maximize f=4x-y, subject to the constraints
7x+4y\le 28, 2y\le 7, x\ge 0, y\ge 0.



Maximize f=3x+y subject to the constraints
8x+5y\le 40;  4x+3y\ge 12;  x\ge 0 and y\ge 0.



A firm has the cost function C = \dfrac {x^{3}}{3} - 7x^{2} + 111x + 50 and demand function x = 100 - p.
Formulate the total profit function P in terms of x



By graphical method solve the following linear programming problem for maximization.
Objective function Z = 1000 x + 600 y
Constraints x + y \leq 200
4x - y \leq 0
x \geq 20,    x \geq 0, y \geq 0.



A firm has the cost function C = \dfrac {x^{3}}{3} - 7x^{2} + 111x + 50 and demand function x = 100 - p.
Write the total revenue function in terms of x



A shopkeeper sells not more than 30 shirts of each colour. Atleast twice as many white ones are sold as green ones. If the profit on each of the white be Rs. 20 and that of green be Rs. 25; then find out how many of each kind be sold to give him a maximum profit. (Graph need not be drawn)



A sweet-shop makes gift packet of sweets by combining two special types of sweets A and B which weigh 7\ kg. Atleast 3\ kg of A and no more than 5\ kg of B should be used. The shop makes a profit of Rs. 15 on A and Rs. 20 on B per kg. Determine the product mix so as to obtain maximum profit.



a) A manufacturing company makes two models A and B of a product. Each piece of model A requires 9 labor hours for fabricating and 1 labor hour for finishing. Each piece of model B requires 12 labor hours for finishing and 3 labor hours for finishing. For fabricating and finishing, the maximum labor hours available are: 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 12000 on each piece of model B. How many pieces of model A and model B should be manufactured per week to realize a maximum profit? What is the maximum profit per week?
b) Find the value of K so that the function f (x) = \begin{matrix} \begin{cases}Kx+1, & \text{if }x \le 5 \\ 3x-5, & \text{if }x \le 5\end{cases}\end{matrix} at x=5 is a continuous function.



Solve the following L.P.P. by graphical method:

Maximise: Z = 6x + 4y

Subject to x\leq 2, x + y \leq 3, -2x + y \leq 1, x \geq 0, y \geq 0



Solve the following linear programming problem graphically:
Maximize Z = 7x + 10y
subject to the constraints
4x + 6y \leq 240
6x + 3y \leq 240
x \geq 10
x \geq 0, y \geq 0



Consider the linear inequalities
2x+3y\le 6, 2x+y\le 4, x\ge 0, y\ge 0
(a) Mark the feasible region.
(b) Maximise the function z=4x+5y subject to the given constraints.



Solve graphically the linear inequalities
2x+3y\le 7,x+2y\le 4
x\ge 0,y\ge 0
If z=6x+5y is the objective function, find its maximum value.



In a factory, there are two machines A and B producing toys. They respectively produce 60 and 80 units in one hour. A can run a maximum of 10 hours and B a maximum of 7 hours a day. The cost of their running per hour respectively amounts to 2,000 and 2,500 rupees. The total duration of working these machines cannot exceed 12 hours a day. If the total cost cannot exceed Rs. 25,000 per day and the total daily production is at least 800 units, then formulate the problem mathematically.



A furniture shop keeper deals only Teak and Rosewood tables which costs Rs. 25,000 and Rs. 40,000 respectively. His shop can contain a maximum of 250 tables. The profit that he gets on these items are respectively Rs. 4,500 and Rs. 5000.
He plans to invest a maximum of Rs. 70 lakhs in his business.
(i) Formulate this problem.
(ii) Write the objective function if he wishes to make maximum profit.



Define feasible region.



Consider the following L.P.P.
Maximize Z=3x+2y
Subject to the constraints
x+2y\le 10
3x+y\le 15
x,y\ge 0
(a) Draw its feasible region.
(b) Find the corner points of the feasible region.
(c) Find the maximum value of Z.



a) Minimize and maximize Z=x+2y, subject to the constraints 
x+2y \ge 100
2x-y \le 0
2x+y \le 200
x,y \ge 0 by graphical method.
b) Prove that \begin{vmatrix}b+c&a&a\\b&c+a&b\\c&c&a+b\end{vmatrix}=4abc



By Graphical  method solve the following L.P.P under the following constraints.
x+y\le 24,2x+y\le 32,x,y\ge 0. Find the maximum value of z=150x+250y



A toy company manufactures two types of doll; a basic version-doll A and a deluxe version doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum of 2000 per day if it produces only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. If company makes profit of Rs. 3 and Rs. 5 per doll, respectively, on doll A and B; how many each should be produced per day in order to maximize profit.



A farmer has a supply of chemical fertilizer of type A which contains 10\% nitrogen and 6\% phosphoric acid and of type B which contains 5\% nitrogen and 10\% phosphoric acid. After soil test, it is found that at least 7 kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs Rs. 5.00 per kg and the type B costs Rs. 8.00 per kg. Using Linear programming, find how many kilograms of each type of the fertilizer should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph. 



Solve the linear programming problem by graphical method with the following restrictions.
3x+5y \leq 5; 5x+2y\leq 10; x\geq 0, y\geq 0 and maximize Z=5x+3y.



Show the solution of the problem of linear programming under the following restrictions by graphical method: x+y\leq 4, x \geq 0, y\geq 0.



A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmans time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.



(Manufacturing problem) A manufacturer has three machines I, II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.
The number of hours required for producing 1 unit of each of M and N on the three machines are given in the following table:
ItemsNumber of hours required on machines
IIIIII
M121
N211.25
She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?



One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes that can be made from 5kg of flour and 1\ kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.



(Manufacturing problem) A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 12000 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?



(Allocation problem) A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated at Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?



Solve the following Linear Programming Problems graphically:
Minimise Z = -3x + 4y
subject to x + 2y\leq 8, 3x + 2y \leq 12, x\geq 0, y \geq 0.



Solve the following Linear Programming Problems graphically:
Maximise Z = 5x + 3y
subject to 3x + 5y\leq 15, 5x + 2y \leq 10, x \geq 0, y \geq 0.



Solve the following Linear Programming Problems graphically :
Minimize Z = 3x + 5y
subject to x + y \leq 4, x  \geq 0 and y \ge 0



Solve the following linear programming problem graphically :
Minimise Z = 200 x + 500 y ... (1)
subject to the constraints:
x + 2y \geq 10 ... (2)
3x + 4y \leq 24 ... (3)
x \geq 0, y \geq 0 ... (4).



A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:
Types of ToysMachinesMachinesMachines
IIIIII
A12186
B609
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs. 7.50 and that on each toy of type B is Rs. 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.  



Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:
Transportation cost per quintal (in Rs)
From/ ToAB
D64
E32
F2.503
How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?



A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?



A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:
FoodVitamin AVitamin BVitamin C
X123
Y221
One kg of food X costs Rs. 16 and one kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet?



There are two types of fertilisers F_{1} and F_{2}\cdot F_{1} consists of 10% nitrogen and 6%phosphoric acid and F_{2} consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14\ kg of nitrogen and 14\ kg of phosphoric acid for her crop. If F_{1} costs Rs. 6/kg and F_{2} costs Rs. 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?



Use the graphical method to solve each of the following LP problems.
LESCO Engineering produces chairs and tables. Each table takes four hours of labour from the carpentry department and two hours of labour from the finishing department. Each chair requires three hours of carpentry and one hour of finishing. 
During the current week, 240 hours of carpentry time are available and 100 hours of finishing time. Each table produced gives a profit of E70 and each chair a profit of E50. How many chairs and tables should be made in order to maximize profit?



If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?



A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs. 5 and that from a shade is Rs. 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?



An oil company has two depots A and B with capacities of 7000\ L and 4000\ L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distances (in km) between the depots and the petrol pumps is given in the following table:
Distance in (km.)
From/ ToAB
D73
E64
F32
Assuming that the transportation cost of 10 litres of oil is Re. 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum?
What is the minimum cost?



A fruit grower can use two types of fertilizer in his garden, brand P and brand Q.The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240\ kg of phosphoric acid, at least 270\ kg of potash and at most 310\ kg of chlorine.
If the grower wants to minimise the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?
Kg per bag
Brand PBrand Q
Nitrogen33.5
Phosphoric acid12
Potash31.5
Chlorine1.52



Holiday Mean Turkey Ranch is considering buying two different types of turkey feed. Each feed contains, in varying proportions, some or all of the three nutritional ingredients essential for fattening turkeys. Brand Y feed costs the ranch $0.2 per pound. Brand Z costs $.03 per pound. The rancher would like to determine the lowest-cost diet that meets the minimum monthly intake requirement for each nutritional ingredient.
The following table contains relevant information about the composition of brand Y and brand Z feeds, as well as the minimum monthly requirement for each nutritional ingredient per turkey.
Composition of Each Pound of Feed
IngredientBrand Y FeedBrand Z FeedMinimum Monthly Requirement
A5 oz10 oz90 oz
B4 oz3 oz48 oz
C.5 oz01.5 oz
Cost/lb $0.2 $.03



A farmer plans to mix two types of food to make a mix of low cost feed for the animals in his farm. A bag of food A costs $10 and contains 40 units of proteins, 20 units of minerals and 10 units of vitamins. A bag of food B costs $12 contains 30 units of proteins, 20 units of minerals and 30 units of vitamins. How many bags of food A and B should the consumed by the animals each day in order to meet the minimum daily requirements of 150 units of proteins, 90 units of minerals and 60 units of vitamins at a minimum cost?



A store sells two types of toys, A and B. The store owner pays $8 and $14 for each one unit of toy A and B respectively. One unit of toys A yields a profit of $2 while a unit of toys B yields a profit of $2. The store owner estimates that no more than 2000 toys will be sold every month and he does not plan to invest more than $20,000 in inventory of these toys. How many units each type of toys should be stocked in order to maximize his monthly total profit?



Solve using graphical method:
Maximize: Z=4x+5y
Subject to:
2x+5y\le 25
6x+5y\le 45
49
x\ge 0, y\ge 0



Solve the following problem:
Maximize Z=-P{x}_{1}+4{x}_{2}
Subject to: -3{x}_{1}+{x}_{2}\le 6
{x}_{1}+2{x}_{2}\le 4
{x}_{2}\le -3
No lower bound constraint for {x}_{1}



Suppose a manufacturer of printed circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits.
Type A requires 20 resistors, 10 transistors and 10 capacitors
Type B requires 10 resistors, 20 transistors and 30 capacitors.
If the profit on type A circuits is E5 and that on type B circuits is E12, how many of each circuit should be produced in order to maximize profit?



A company produces two types of tables, {T}_{1} and {T}_{2}. It takes 2 hours to produce the parts of one unit of {T}_{1}, 1 hour to assemble and 2 hours ot polish. It takes 4 hours to produce the parts of one unit of {T}_{2}, 2.5 hour to assemble and 1.5 hours of polish. Per month, 7000 hours are available for producing the parts, 4000 hours for assembling the parts and 5500 hours for polishing the tables. The profit per unit of {T}_{1} is $90 and per unit of {T}_{2} is $110. How many of each type of tables should be produced in order to maximize the total monthly profit?



Use the graphical method to solve each of the following LP problems.
A factory manufactures two products, each requiring the use of three machines. The first machine can be used at most 70 hours; the second machine at most 40 hours; and the third machine at most 90 hours. The first product requires 2 hours on Machine 1, 1 hour on Machine 2, and 1 hour on Machine 3; the second product requires 1 hour each on machine 1 and 2 and 3 hours on MachineIf the profit in E40 per unit for the first product and E60 per unit for the second product, how many units of each product should be manufactured to maximize profit?



Using graphical method
Find the corner points for
2x+5y\le 25
6x+5y\le 45
x\ge 0,y\ge 0



Use graphical method
Minimize: z=600x+900y
Subject to:
40x+60y\ge 480
30x+15y\ge 180
x\ge 0,y\ge 0



Construct the graphs of the following functions.
\displaystyle y \, = \, x \, | x | \, - \, 4x \, - \, 5.



Construct the graphs of the following functions.
\displaystyle y \, = \, (2x \, + \, 1) \, / \, (x \, - \, 1).



Construct the graphs of the following functions.
\displaystyle y \, = \, ( |1 \, - \, x \, | \, + \, 2) \, (x \, + \, 1).



Construct the graphs of the following functions.
\displaystyle y \, = \,  | 2x^2 \, - \, 3x \, + \, | x \, - \, 1| \, |.



Construct the graphs of the following functions.
\displaystyle y \, = \,  1 / | x |.



Construct the graphs of the following functions.
\displaystyle y \, = \,  \frac{| x \, - \, 1 |}{x \, - \, 1} \, (x^2 \, + \, 3).



John has $20,000 to invest in three funds {F}_{1},{F}_{2} and {F}_{3}. Fund {F}_{1} is offers a return of 2% and has a low risk. Fund {F}_{2} offers a return of 4% and has a medium risk. Fund {F}_{3} offers a return of 5% but has a high risk. To be the safe side, John invests no more than $3000 in {F}_{3} and at least twice as much as in {F}_{1} than in {F}_{2}. Assuming that the rates hold till the end of the year, what amounts should he invest in each fund in order to maximize the year end return?



Construct the graphs of the following functions.
\displaystyle y \, = \,  x (x \, - \, 3)^2.



Each month a store owner can spend at most $100,000 on PC's laptops. A PC costs the store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while laptop is sold for a profit of $700. The store owner estimates that at least 15 PC's but no more than 80 are sold each month. He also estimates that the number of laptops sold is at most half the PC's. How many PC's and how many laptops should be sold in order to maximize the profit?



Solve the following maximal assignment problem:
Branch ManagerMonthly Business
(Rs. lakh)
ABCD
P111199
Q13161110
R1217138
S16141612



Mr. Anil wants to invest at most Rs. 60,000 in Fixed Deposit (F.D.) and Public Provident Fund (P.P.F.). He wants to invest at least Rs. 20,000 in F.D. and at least Rs. 15,000 in P.P.F. The rate of interest on F.D. is 8\% p.a. and that on P.P.F. is 10\% p.a. Formulate the above problem as L.P.P. to determine maximum yearly income.



Minimize and maximize Z=600x+400y
Subject to the constraints
x+2y\le 12
2x+y\le 12
4x+5y\le 20
x\ge 0;y\ge 0 by graphical method



Solve the following using graphical method:
Minimize: Z = 3x + 5y subject to 2x+ 3y \geq 12, -x+y \leq 3, x \leq 4, \,\, y \geq 3, \,\, x \geq 0, \,\, y \geq 0



Solve the following minimal assignment problem and hence find minimum time where '-' indicates that job cannot be assigned to the machine:
MachinesProcessing time in hours
ABCDE
M_{1}911151011
M_{2}129-109
M_{3}-1114117
M_{4}1481278



Solve the following problem graphically:
Minimise and Maximise
z = 3x + 9y
Subject to the constraints:
x + 3y \leq 60
x + y \geq 10
x \leq y
x\geq 0, y\geq 0.



Construct the graphs of the following functions.
\displaystyle y \, = \, (2 | x | -1) \, / \, (x \, - \, 3).



In solving LP problem : minimize z = 6x + 10y, subject to x \geq 6, y \geq 2, 2x + y \geq 10, x \geq 0, y \geq 0, which constraints are reduntant?



Minimise Z= 3x +2y
subject to constraints: 
x+y \geq 8
3x+ 5y\leq 15
x\geq 0, y \geq 0



A chemical company produces two products,X and Y.Each unit of product X requires 3hrs on operation I and 4hrs on operation II,while each unit of product Y requires 4hrs on operation I and 5hrs on operationII.
Total available time for operations I and II is 20hrs and 26 hrs respectively.The production of each unit of product Y also results in two units of a by-product Z at no extra cost.
Product X sells at profit of Rs.10/unit, while Y sells at profit of Rs.20/unit.By-product Z brings a unit profit of Rs.6 if sold;in case it cannot be sold, the destruction cost is Rs.4 per unit.Forecasts indicate that not more that 5 units of Z can be sold.Formulate the L.P model to determine the quantities of X and Y to be produced,keeping Z in mind, so that the profit earned is maximum.
  



An aeroplane can carry a maximum of 250 passengers, A profit of Rs. 1500 is made on each executive class ticket and a profit of Rs. 900 is made on each economy class ticket. The airline reserves at least 30 seats for executive class. However at least 4 times as many passengers prefer to travel by economy class than by executive class. Formulate LPP in order to maximize the profit For the airline.



One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.



 A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are given in the table below:
MachineTime per unit
(in minutes)
of product-1
Time per unit
(in minutes)
of product-2
Time per unit
(in minutes)
of product-3
Machine capacity
(minutes/day)
{M}_{1}232440
{M}_{2}4-3470
{M}_{2}25-430
It is required to determine the daily number of units to be manufactured for each product. The profit per unit for product 1,2 and 3 is Rs.4, Rs.3 and Rs.6 respectively. It is assumed that all the amounts produced are consumed in the market. Formulate the mathematical(L.P) model that will maximize the daily profit.



A firm produces an alloy having the following specifications:
\left(i\right) specific gravity\le0.98
\left(ii\right)chromium\ge8percent
\left(iii\right)melting point\ge{450}^{0}C
Raw materials A,B and C having the properties shown in the table can be used to make the alloy.
PropertyProperty of raw
 material A
Property of raw
 material B
Property of raw
material C
Specific gravity0.920.971.04
Chromium7 percent13 percent16 percent
Melting point{440}^{0} Celsius{490}^{0} Celsius{480}^{0} Celsius
Costs of the various raw materials per ton are:Rs.90 for A,Rs.280 for B and Rs.40 for C. Formulate the L.P model to find the proportions in which A,B and C be used to obtain an alloy of desired properties while the cost of raw materials is minimum.



A company has two grades of inspectors,I and II to undertake quality control inspection.At least 1,500 pieces must be inspected in an 8-hour day.GradeI inspector can check 20 pieces in an hour with an accuracy of 96percent.
GradeII inspector checks 14 pieces an hour with an accuracy of 92 percent.
Wages of grade I inspector are Rs.5 per hour while those of grade II inspector are Rs.4 per hour. Any error made by an inspector costs Rs.3 to the company.If there are,in all, 10  grade I inspectors and 15 grade II inspectors in the company,find the optimal assignment of inspectors that minimizes the daily inspection cost. 



A person wants to decide the constituents of a diet which will fulfill his daily requirements of proteins,fats and carbohydrates at the minimum cost.The choice is to be made from four different types of foods.The yields per unit of these foods are given in table below.
Food TypeProteins
(yield per unit)
Fats
(yield per unit)
Carbohydrates
(yield per unit)
Cost per unit
(in Rupees)
132645
242440
387785
465465
Minimum
Requirement
800200700
Formulate the linear programming model for the problem.



Solve the following system of inequalities graphically 2x+y\ge 6, 3x+4y\le 12



Maximize Z=x+2y subject to 2x+y\ge 3,x+2y\ge 6, x,y\ge 0.



Draw the line 2y=3x+4 on the xy plane.



Solve the following system of inequalities graphically x+y\ge 4, 2x-y> 12



Solve the following system of inequalities graphically x+y\le 9, y>x, x\ge 0



Solve the following system of inequalities graphically 5x+4y\le 20, x\ge 1, y\ge 2



Solve the following system of inequalities graphically 2x-y>1, x-2y<-1



Solve the following system of inequalities graphically 2x+y\ge 8, x+2y\ge 10



Solve the following system of inequalities graphically x+y\le 6, x+y\ge 4



Ravish tells his daughter Aarushi, "Seven years ago, I was seven times as old as you were then. Also, there years from now, I shall be three times as old as you will be. If the present ages of Aarushi and Ravish are x and y years respectively, represent this situation algebraically as well as graphically.



Solve the following system of inequalities graphically 3x+4y\le 60, x+3y\le 30, x\ge 0, y\ge 0



If the point (2K-3,K+2) lies on the graph of equation 2x+3y+15=0, find the value of k.



Solve the following Linear programming problems graphically :
Maximizez = 3x + 4y subject to the constraints : x + y \le 4, \, x \ge 0 , \, y \ge 0



See the given figure and write the following:
(i) The coordinates of B
(ii) The coordinates of C
(iii) The point identified by the coordinates (-3,-5)
(iv) The ordinate of the point B.
1068614_90cf0140a4e94c19a010adcc9570c106.png



Maximize Z=-x+2y, subject to the constraints: x\geq 3; x+y\geq 5, x+2 y\geq 6, y\geq 0.



Solve the following system of inequalities graphically 2x+y\ge 4, x+y\le 3, 2x-3y\le 6



Solve the following Linear Programming problems graphically:
Maximize Z = 3x + 4y       Subject to the constraints : x + y \le 4, \, x \ge 0 , \, y \ge 0
Minimize Z = -3x + 4 y       subject to x + 2y \le 8 , \, 3x + 2y \le 12, \, x \ge 0 , \, y \ge 0



x-y\leq -1,x+y\leq 0,x,y\geq0



If a 20 year old girl drives her car at 25\ km/h, she has to spend Rs\ 4/km on petrol. If she drives her car at 40km/h, the petrol cost increases to Rs\ 5\ /km. She has Rs\ 200 to spend on petrol and wishes to find the maximum distance she can travel within one hour. Express the above problem as Linear Programming Problem. Write any one value reflected in the problem.



A store sells two types of toys, A and B. The store owner pays Rs. 8 and Rs. 14 for each unit of toy A and toy B respectively. One unit of toy A yields a profit of Rs. 2 while a unit of toy B yields a profit of Rs.The store owner estimates that no more than 2000 toys will be sold every month and he does not plan to invest more than Rs. 20,000 in inventory of these toys. How many units of each type of toys should be stocked in order to maximize his monthly total profit ?



Find the solution of the following LP problem ''Maximize z=x+ y subject x + y \leq 1,-3x+y\geq 3,x\geq  0,y\geq 0'' is.............



A  firm is engaged in producing two models {X_{1.}}{X_2}  performing only three operations assembling, pianting and testing.The relevant data are as follows.
Total number of hours available each week for assemblling 600 hours, painting 100 hours and testing 30 hours.The firm wishes to determine its weekly products mix so as to maximize revenue formulate I.P.P model for maximize the revenue
1199695_822f91663e4d438f8a98650c0d279755.png



There are 50 apples trees in an orchard. Each tree produces 800 apples. For each additional tree planeted in the orchord. The output additional tree drops by 10 apples tree drops by10 apples. Number of trees that should to the existing orchord for maximising the output of the trees is 3k, then k= ?



A producer has 20 and 10 units of labour and capital respectively which he can use to produce two kinds of goods X and Y. To produce one unit of X,2 units of capital and 1 unit of labour is required. To produce one unit of Y,3 units of labour and 1 unit of capital is required. Find X and Y ?



Maximize z=3x+5y, Subject to x+4y \le 24, 3x+y \le 21,x+y \le 9,x \ge 0. Also find maximum value of z.



A small firm makes necklaces and bracelets. The combined number of necklaces and bracelets that it can handle per day is at most 24. A bracelet takes one hour to make and necklace takes 30 minutes. The maximum number of hours available per day is 16. If the profit on a bracelet is Rs. 300/- and on a necklace is Rs. 100/-, How many necklaces and bracelets must be made per day to get the maximum profit? Make it as an LLP and solve graphically.



Maximize z = 2x + 4y subject to the constraints

x ≥ 0, y ≥ 0, x + y ≥ 1, 3x + 2y <= 6



Maximise ,Z=x+y, subject to x-y\leq -1-x+y\leq 0x,y\leq 0



A library has to accommodate two different types of books on a shelf. The books are 6cm and 4cm thick and weigh 1kg and 1.5kg respectively. The shelf is 96cm long and at most can support a weight of 21kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LLP and solve it graphically.



A small firm manufactures gold rings and chains.The total number of rings and chains manufactured per day is almost 24. It takes 1\ h to make a ring and 30 min to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, then find the number of rings and chains that then find the number of rings and chains that should be manufactured per day so as to earn the maximum profit. Make it as an LPP and solve it graphically.



The standard weight of a special purpose brick is  5\mathrm { kg }  and It must contain two basic ingredients   B _ { 1 } \text { and } B _ { 2 } , B _ { 1 }  costs  Rs. 5  per kg and  B _ { 2 }  costs  Rs.8  per kg, Strength considerations dictate that the brick should contain not more than  4\mathrm { kg }  of  B _ { 1 }   and   minimum  2 \mathrm { kg }  of  \mathrm { B } _ { 2 } .  Since the demand for the product is likely to be related to the price of the brick. find the minimum cost of brick satisfying the above conditions, Formulate this situation as an  LPP  and solve it graphically.



A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3g of silver and 1g of gold while that of B requires 1g of silver and 2g of gold. The company can use atmost 9g of silver and 8g of gold. If each unit of type A brings a profit of 40 and that of type B 50, find the number of units of each type that the company should produce to maximize the profit. Formulate and solve graphically the LPP and find the maximum profit.



A man has Rs. 1500 for purchasing wheat and rice. A bag of rice and a bag of wheat cost Rs.  180 and Rs. 120 respectively. He has a storage capacity of only 10 bags. He earns a profit of Rs. 11 and Rs. 9 per bag of rice and wheat, respectively. Formulate the problem as an LPP to find the number of bags of each type he should buy for getting maximum profit and solve it graphically.



Maximize Z=12x+24y subject to the constraints x+y\ge 5, 5x+7y\le 35, x-y\ge0,x,y\ge0 graphically.



A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs. 400 and each small van is Rs.Not more than Rs. 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.



One kind of cake requires 300gm of flour and 15gm of fat and another kind of cake requires 150gm  of flour and  30gm  of fat. Find the maximum number of cake that can be made from  7.5kg  of flour and  600gm  of fat . Form a linear programming problem and solve it graphically.



A manufacturer of patent medicines is preparing a production plan on medicines A and B. There are sufficient row material available to make 20000 bottles of A and 40000 bottles of B, but there are 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is Rs. 8 per bottle for A and Rs. 7 per bottle for B. Construct the maximization problem.



Maximize z=6x+4y, Subject to x\leq 2, x+y\leq 3, -2x+y\leq 1, x\geq 0, y\geq 0. Also find maximum value of z.



A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and  8 minutes each for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours available for assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit ? Formulate the above LPP and solve it graphically and also find the maximum profit.



If the replacement set is the set of all non-negative integers, find the solution set of the following 
A) x + 1 < 9
B) 3x - 4 \leqslant 20
C) 5 + 2x > 10
D) 10 < x + 5 < 21



An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitation, shop A has 180 man-days per week available whole shop B has 135 man-days per week. If the manufacturer makes a profit of Rs. 30000 on each truck and Rs. 2000 on each automobile, how many of each should he produce to maximum his profit? Formulate this as a LPP.



A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of A and B, and the number of man-hours the firm has available per week are as follows:
GadgetFoundryMachine-shop
A105
B64
Firm's capacity per week1000600
The profit of the sale of A is Rs. 30 per unit as compared with Rs. .20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.



Two tailors A and B earn Rs. 300 and Rs. 400 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.



Vitamin A and B are found in two different foods F_{1} and F_{2}. One unit of food F_{1} contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F_{2} contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F_{1} and F_{2} cost  Rs. 50 and 25 respectively. The minimum daily requirement for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any things in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F_{1} and F_{2} at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.



Solve each of the following linear programming problem by graphical method.
Minimize Z=5x+3y
Subject to 2x+y\ge 10
                 x+3y\ge 15
                   x\le 10
                   y\le 8
                  x, y\ge 0



Solve each of the following linear programming problem by graphical method.
Minimize Z=18x+10y
Subject to 4x+y\le 20
                 2x+3y\le 30
                   x,y\ge 0



Solve each of the following linear programming problem by graphical method.
Maximize Z=7x+10y
Subject to x+y\le 30000
                   y\le 12000
                    x\ge 6000
                      x\ge y
                     x, y\ge 0



Solve each of the following linear programming problem by graphical method.
Minimize Z=2x+4y
Subject to x+y\ge 8
                 x+4y\ge 12
                 x\ge 3, y\ge 2



A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large and the number of large vans is Rs. 400 and each small can is Rs. 200. Not more than Rs. 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.



Solve each of the following linear programming problem by graphical method.
Maximize Z=15x+10y
Subject to 3x+2y\le 80
                 2x+3y\le 70
                   x,y\ge 0



Solve each of the following linear programming problem by graphical method.
Maximize Z=10x+6y
Subject to 3x+y\le 12
                 2x+5y\le 34
                   x,y\ge 0



Solve each of the following linear programming problem by graphical method.
Maximize Z=9x+3y
Subject to 2x+3y\le 13
                 3x+y\le 5
                   x,y\ge 10



Solve each of the following linear programming problem by graphical method.
Maximize Z=3x+4y
Subject to 2x+2y\le 80
                 2x+4y\le 120



Solve each of the following linear programming problem by graphical method.
Maximize Z=4x+3y
Subject to 3x+4y\le 24
                 8x+6y\le 48
                   x\le 5
                   y\le 6
                  x, y\ge 0



A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 of calories. Two foods A and B, are available at a cost of Rs. 5 and Rs. 4 per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the food B contains 100 unit of vitamins, 2 unit of minerals and 40 units of calories. Find what combination of foods should be used to have the least cost, but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically.



Kellogg is a new cereal formed of a mixture of a mixture of bran and rice that contains at least 88 grams of proteins and at least 36 milligrams of iron. knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice constant 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kg and rice costs Rs. 4 per kg.



A publisher sells a hard cover edition of a text book for Rs. 72 and a paperback edition of the same ext for Rs. 40. Costs to the publisher are Rs. 56 and Rs. 28 per book respectively in addition to weekly costs of Rs. 9600. Both types requires 5 minutes of printing time, although hardcover requires 10 minutes binding time and the paperback requires only 2 minutes. Both the printing the binding operation have 4,800 minutes available each week. How many of each type of book should be produced in order to maximize profit?



A gardener has supply to fertilizer of type I which consists of 10\% nitrogen and 6\% phosphoric acid and type II fertilizer which consists of 5\% nitrogen and 10\% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14 kg of phosporic acid for his crop. If the type I fertilizer costs 60 paise per kg and type II fertilizer costs 40 paise per kg, determine how many kilogram of each fertilizer should be used so that nutrient requirement are met at a minimum cost. What is the minimum cost?



Anil wants to invest at most Rs. 12000 in saving certificates and national saving bonds. According to rule, he has to invest at least Rs. 2000 in saving certificates and at least Rs. 4000 in National Saving Bonds. If the rate of interest on saving certification is 8\% per annum and the rate of interest on National Saving Bond is 10\% per annum, how much money should he invest to earn maximum yearly income? Also find his maximum yearly income.



Solve each of the following linear programming problem by graphical method.
Maximize Z=3x+5y
Subject to x+2y\le 20
                 x+y\le 15
                   y\le 5
                   x,y\ge 0



A firm manufactures two types of products A and B and sells them at a profit of Rs. 5 per unit of type A and Rs. 3\ per unit of type B. Each product is processed on two machines M_{1} and M_{2}. One unit of type A requires one minutes of processing time on M_{1} and two minutes of processing time on M_{2}, whereas one unit of type B requires one minutes of processing time on M_{1} and one minutes on M_{2}. Machines M_{1} and M_{2} are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product should the firm produce a day in order to maximize the profit. Solve the problem graphically.



A company manufactures two types of types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 hours minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs. 50 each on type A and Rs. 60 each on type B. How many toys of each type should the company manufacture in a day to maximize  the profit?



A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A,7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1\ kg of food X and 1\ kg of food Y are given below:
Vitamin AVitamin BVitamin CVitamin D
Food X1112
Food Y2131
One kg of food X costs Rs\ 5, whereas one kg of food Y costs Rs\ 8. Find the least cost of the mixture which will produce the desired diet.



A manufacturer produces two types of steel trunks requires 3 hours on machine A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B. Machine A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit if Rs. 30 and Rs. 25 per trunk of the first type and the second type respectively. How many trunks of each type must he make each day to make maximum profit?



There are two types of fertilizers 'A' and 'B'. 'A' consists of 12\% nitrogen and 5\% phosphoric acid whereas 'B' consists of 4\% nitrogen and 5\% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acis for his crops. If 'A' costs Rs. 10 per kg and 'B' cost Rs. 8 per kg, then graphically determine how much of each type of fertilizers should be used sot hat nutrient requirements are met at a minimum cost.



A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman's time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works a full capacity.



Find the maximum value of Z = 7x + 7y, subject to the constraints x\geq 0, y \geq, x + y \geq 2 and 2x + 3y \leq 6.



Solve the following linear programming problem:
Maximise: z=150x+250y
Subject to: 4x+y \le 40
                  3x+2y\le 60
                  x\ge 0
                  y\ge 0



Show that the solution set of the following linear constraints is empty:
x - 2y \geq 0, 2x - y \leq -2, x \geq 0 and y\geq 0.



A company manufactures two articles A and B. There are two department through which these article are processed : (i) assembly and (ii) finishing department. The maximum capacity of the first department is 60 hours a week and that of other department. 48 hours, per week. The product of each unit of article A requires 4 hours in assembly and 2 hours in finishing and that of each unit of B requires 2 hours in assembly and 4 hours in finishing. If the profit is Rs. 6 for each unit A and Rs. 8 for each unit of A and Rs. 8 for each unit of B, find the number of units of A and B to be produced per week in order to have maximum profit.



A medical company has factories at two places, A and B. From these places, supply is made to each of its three agencies situated at P,Q and R. The monthly requirement of the agencies are respectively 40, 40 and 50 packets of the medicines, while the production capacity of the factories, A and B are 60 and 70 packets respectively. The transportation cost per packet from the factories to the agencies are given below:
Transportation cost per packet (in Rs)
To \downarrow \ From \toAB
P54
Q42
R35
How many packets from each factory be transported to each agency so that the cost of transportation is minimum cost?



Find the linear constraints for which the shaded area in the figure given is the solution set.
1708377_211fb210d06d4fed8d58d9f252cffd41.png



A man owns a field of area 1000\ m^{2}. He wants to plant fruit trees in it. He has a sum of Rs. 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10\ m^{2} of ground per tree and costs Rs. 20 per tree, and type B requires 20\ m^{2} of ground per tree and costs Rs. 25 per tree. When full grown, a type-A tree produces and average of 20\ kg of fruit which can be sold at a profit of Rs. 2 per kg and a type -B tree produces an average of 40\ kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when trees are fully grown? What is the maximum profit?



A manufacturer produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of Rs. 0.25 and Rs. 0.50 each. Assuming that the manufacturer can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?



Maximize Z = 3x + 5y, subject to the constraints x + 2y \leq 2000, x + y \leq 1500, y \leq 600, x \geq 0 and y\geq 0.



Mr Dass wants to invest Rs. 12000 in Public Provident Fund (PPF) and in National bonds. He has to invest at least Rs. 1000 in PPF and at least Rs. 2000 in bonds. If the rate of interest on PPF is 12\% per annum and that on bonds is 15\% per annum, how should he invest the money to earn maximum annual income? Also find maximum annual income.



Maximize Z = 4x + 9y, subject to the constraints x\geq 0, y \geq 0, x + 4y \leq 200, 2x + 3y \leq 134.



Minimize Z = 2x + 3y, subject to the constraints x \geq 0, y \geq 0, x + 2y \geq 1 and x + 2y \leq 10.



A company producing soft drinks has a contract which requires a minimum of 80 units of chemical A and 60 units of chemical B to go in each bottle of the drink. The chemicals are available in a prepared mix from two different suppliers. Supplier X has a mix of 4 units of A and 2 units of B that costs Rs. 10, and the supplier Y has a mix of 1 unit of A and 1 unit of B that costs Rs. 4. How many mixes from X and Y should the company purchase to honour the contract requirement and yet minimize the cost?



A firm manufactures two types of products, A and B, and sells them at a profit of Rs. 2 on type A and Rs. 2 on type B. Each product is processed on two machines, M_{1} and M_{2}. Type A requires one minute of processing time on M_{1} and two minutes on M_{2}. Type B requires one minute on M_{1} and one minute on M_{2} is available for at most 10 hours a day.
Find how many producs of each type the firm should produce each day in order to get maximum profit.



Find the minimum value of Z = 3x + 5y, subject to the constraints -2x + y \leq 4, x + y \geq 3, x - 2y \leq 2, x \geq 0 and y\geq 0.



Find the maximum and minimum values of Z = 2x + y. subject to the constraints
x + 3y \geq 6, x - 3y \leq 3, 3x + 4y \leq 24, -3x + 2y \leq 6, 5x + y \geq 5, x \geq 0 and y \geq 0.



An aeroplane can carry a maximum of 200 passengers. A profit of 500 \ Rs. is made on each executive class ticket out of which 20 \% will go to the welfare fund of the employees. Similarly a profit of 400 \ Rs. is made on each economy ticket out of which 25 \% will go for the improvement of facilities provided to economy class passengers. In both cases, the remaining profit goes to the airline's fund. The airline reserves at least 20 seats for executive class. However, at least four times as many passengers prefer to travel by economy class than by the executive class. Determine, how many tickets of each type must be sold in order to maximise the net profit of the airline. Make the above as an LPP and solve graphically.



A brick manufacturer has two depots, P and Q, with stocks of 30000 and 20000 bricks respectively. He receives order from three buildings A, B, C for 15000, 20000 and 15000 bricks respectively. The cost of transporting 1000 bricks to the buildings from the depots are given below.
Cost of transportation
(in Rs. per 1000 bricks)
From ToABC
P402030
Q206040
How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?



Maximise z = 8x + 9y subject to the constraints given below :
2x + 3y \leq 6; 3x - 2y \leq 6; y \leq 1; x\geq 0; y \geq 0



A manufacturer has 640 litres of a 8\% solution of boric acid. How many litres of a 2\% boric acid solution be added to it so that the boric acid content in the resulting mixture will be more than 4\% but less than 6\%?



A company manufactures cassettes. Its cost and revenue functions are C(x)=26000+30x and R(x)=43x respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?



How many litres of water will have to be added to 600 litres of the 45\% solution of acid so that the resulting mixture will contain more than 25\%, but less than 30\% acid content?



The water acidity in a pool is considered normal when the average pH reading of three daily measurements between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of the pH values for the third reading that will result in the acidity level being normal.



A village has 500 hectare of land to grow two types of plants, X and Y. The contributions of total amount of oxygen produced by plant X and plant Y are 60 \% and 40 \% per hectare respectively. To control weeds, a liquid herbicide has to be used for X and Y at rates of 20 litres and 10 litres per hectare, respectively. Further no more than 8000 litres of herbicides should be used in order to protect aquatic animals in a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total production of oxygen?



A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. They need certain nutrients, names as X, Y, Z. The pigs are fed on two products, A and B. One unit of product A contains 36 units of X, 3 units of Y and 20 units of Z, while one unit of product B contains 6 units of X, 12 units of Y and 10 units of Z. The minimum requirements of X, Y, Z are 108\ units, 36\ units and 100\ units. respectively. Product A costs Rs. 20 per unit and product B costs Rs. 40 per unit. How many uits of each product must be taken to minimize the cost? Also, find the minimum cost.



Determine the maximum value of Z = 3x + 4y if the feasible region (shaded) for a LPP is shown in Fig. 
1799026_85bd0052140042df9d4e934474bef8d7.png



The feasible region for a LPP is shown in Fig. 12.Find the minimum value of Z = 11x + 7y.
1799058_af685bd3e82c4f56ad8f2723370b10d6.PNG



Feasible region (shaded) for a LPP is shown in Fig., Maximise Z = 5x + 7y.
1799036_b83cf270635643a2ad62e8b2cbb53af3.png



Maximise Z = 3x + 4y, subject to the constraints : x + y \leq 1, x \geq 0, y \geq 0.



Minimise Z = 13x - 15y subject to the constraints :
x + y  \leq 7, 2x - 3y + 6 \geq 0, x \geq 0, y \geq 0.



Maximise the function Z = 11x + 7y, subject to the constraints:
x \leq 3, y \leq 2, x \geq 0, y \geq 0



The feasible region for a LPP is shown in Fig. Evaluate Z = 4x + y at each of the corner points of this region. Find the minimum value of Z, if it exists.
1799064_469f37650a6b492aab11ab323b787b00.png



A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs. 50 and that on type B circuit is Rs. 60, formulate this problem as a LPP so that the manufacturer can maximise his profit.



Determine the maximum value of Z = 11x + 7y subject to the constraints : 2x + y \leq 6, x \leq 2, x \geq 0, y \geq 0.



In Fig., the feasible region (shaded) for a LPP is shown. Determine the maximum and minimum value of Z = x + 2y.
1799189_1e69824a5bff4e85a2fe5827086dc571.png



A company manufactures two types of screw A and B. All the screws have to pass through a threading machine and a slotting machine. A box type A screws requires 2 minutes on the threading machine and 3 minutes on the slotting machine. A box of type B screws requires 8 minutes of threading on the threading machine and 2 minutes on the slotting machine. In a week, each machine is available for 60 hours.
On selling these screws, the company gets a profit of Rs 100 per box on type A screws and Rs 170 per box on type B screws.
Formulate this problem as a LPP given that the objective is to maximise profit.



Fill in the blank.
If the feasible region for a LPP is _______, then the optimal value of the objective function Z = ax + by may or may not exist.



Fill in the blank.
In a LPP, the objective function is always _________ .



A man rides his motorcycle at the speed of 50 km/hour. He has to spend Rs 2 per km on petrol. If he rides it at a faster speed of 80 km/hour, the pertrol cost increases to Rs 3 per km. He has at most Rs 120 to spend on petrol one hour time. He wishes to find the maximum distance that he can travel.
Express this problem as a linear programming problem.



A company manufactures two types of sweaters: type A and type B. It costs Rs 360 to make a type A sweater and Rs 120 to make a type B sweater. The company can make at most 300 sweaters and spend at most Rs 72000 a day. The number of sweaters of type B cannot exceed the number of sweaters of type A  by more thanThe company makes a profit of Rs 200 for each sweater of type A and Rs 120 for every sweater of type B. Formulate this problem as a LPP to maximise the profit of the company.



Fill in the blank.
A corner point of a feasible region is a point in the region which is the _________ of two boundary lines.



Refer to ExerciseHow many of circuits of Type A and Type B, should be produced by the manufacturer so as to maximise his profit? Determine the maximum profit.



Fill in the blank.
A feasible region of a system of linear inequalities is said to be ________ if it can be enclosed within a circle.



Fill in the blank.
In a LPP if the objective function Z = ax + by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same ________ value.



Fill in the blank.
The feasible region for an LPP is always a _________ polygon.



Refer to ExerciseWhat will be the minimum cost?



A company makes 3 model of calculators : A, B and C at factory I and factory II. The company has ordered for at least 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 calculators of model B and 30 calculators of model C are made every day; at Factory II, 40 calculators of model A, 20 calculators of model B and 40 calculators of model C are made everyday. It costs Rs 12000 and Rs 15000 each day to operate factory I and II, respectively. Find the number of days each factory should operate to minimise the operating costs and still meet the demand.



Refer to ExerciseHow many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit?



 Tablets IronCalcium Vitamin 
X
Y
6
2
3
3
2
4
In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligram per tablet) are given as below.

The person needs at least 18 milligram of iron, 21 milligram of calcium and 16 milligram of vitamins. The price of each tablet of X and Y is Rs 2 and Re 1 respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?



In a cattle breeding firm, it is prescribed that the food ration from one animal must contain 14.22 and 1 units of nutrients A, B, and C respectively. Two different kinds of fodder are available. Each unit of these two contains the following amounts of these three nutrients:
Fodder\rightarrow
NutrientFodder 1Fodder 2
Nutrients A21
Nutrients B
21
Nutrients C11
The cost of fodder 1 is \square 3 per unit and that of fodder 2 \square 2.. Formulate the LPP to minimize the cost .



A manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry and then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for production of A and B and the number of man-hours available for the firm is as follows:
GadgetsFoundryMachine shop
A105
b64
TIme available (Hour)6035
Profit on the sale of A is \square 30 and B is \square 20 \;per \;units. Formulate the LPP to have maximum profit.



Refer to ExerciseSolve the linear programming problem and determine the maximum profit to the manufacturer.



Refer to ExerciseDetermine the maximum distance that the man can travel.



A manufacturer  produces two models of bikes- model X and model Y. Model X takes a 6 man-hours to make per unit, while model Y takes 10 man-hours per unit. There is a total of 450 man-hour available per week. Handling ans marketing costs are Rs 2000 and Rs 1000 per unit for models X and Y respectively. The total funds available for these purpose are Rs 80,000 per week. Profits per unit for models X and Y are Rs 1000 and Rs 500, respectively. How many bikes of each model should the manufacturer produce so  as to yield a maximum profit? Find the maximum profit.



Maximise Z = x + y subject to x + 4y \leq 8, 2x + 3y \leq 12, 3x + y \leq 9, x \geq 0, y \geq 0.



Maximise and minimise Z = 3x - 4y, subject to
x - 2y \leq 0
-3x + y \leq 4
x -y \leq 6
x, y \geq 0



Solve the following LPP by graphical method:
Maximize z = 7x + 11y, subject to 3x + 5y \leq 26, 5x +3y \leq 30, x \geq 0, y \geq 0 



Solve then following LPP graphically:
Maximize : Z = 10x + 25y
Subjec to : x \leq 3, y \leq 3, x + y \leq 5, x \geq 0, y \geq 0.



If john drives a car at a speed of 60 km/hour, he  has to spend \square 5 per km on petrol. If he drives at a faster speed of 90 km/hour, the cost of petrol increases \square 8 per km. he has \square 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as LPP.



A manufacturer produces bulbs and tubes. Each of these must be processed through two machines M_{1} and M_{2}. A package of build requires 1 hour of work on machine M_{1} and 3 hours on machine M_{2}. A package of tubes requires 2 hours on machine M_{1} and 4 hours on machine M_{2}. He earns a profit of \square13.5 per package of bulbs and \square 55 per package of tubes. Formulate the LPP to maximize the profit, if he operates the machine M_{1}, for almost 10 hours a day and machine M_{2} for almost 12 hours a day.



Solve the following LPP by graphical method:
Maximize z = 4x + 6y, subject to 3x + 2y \leq 12, x + y \geq 4, x, y \geq 0.



Solve the following LPP by graphical nethod:
Maximmize : z = 3x + 5y
Subject to : x + 4y \leq 24
                   3x + y \leq 21
                   x + 7 \leq 9
                   x \geq 0, y \geq 0.



A company manufactures two types of fertilizers F_{1} and F_{2}. Each type of fertilizers requires two types of raw materials A and B. The number of unit of A and B required to manufacture one unit of fertilizer F_{1} and F_{2} and availability of the raw materials A and B per day are given in the table below:
Fertilizers\rightarrow
Raw materialF_{1}F_{2}Availability
      A        2      3      40
      B       1      4    70
By selling one unit of F_{1} and one unit of F_{2}, the company gets aprofit of \square500 and \square750 respectively. Formulate the problem as LPP to maximize the profit.



The company makes concrete bricks made up of cement and sand. The weight of a concrete brick has to be at least 5 kg. Cement costs \square20 per kg and sand costs of \square6 per kg. Strength consideration dictates that a concrete brick should contain minimum 4 kg of cement and not more than 2 kg of sand. Form the LPP for the cost to be minimum.



A doctor has prescribed two different units of food A and B to form a weekly diet for a sick person. The minimum requirements of fat, carbohydrates and proteins are 18, 24, 14 units respectively. One unit of food A has 4 units of fat, 14 units of carbohydrates and 8 units of protein. One unit of food B has 6 units of fat, 12 units of carbohydrates and 8 units of protein. The price of food A is \square 4.5 per unit and that of food B is \square 3.5 per unit. Form the LPP, so that the sick person's diet meets the requirements at a minimum cost.



A printing company prints two types of magazines A and B. The company earns \square 10  and \square 15 in magazines A and B per copy. These are processed on three machines I, II, III. Magazine A requires 2 hours on Machine I, 5 hours on Machine II, and 2 hours on Machine III. Magazine B requires 3 hours on Machine I, 2 hours on Machine II, and 6 hours on Machine III. Machines I, II, III are available for 36,50, and 60 hours per week respectively. Formulate the LPP to determine weekly production of magazines A and B, so that the total profit is maximum.



Solve the following LPP:
Maximize z = 4x_{1} + 3x_{2} subject to
3x_{1} + x_{2} \leq 15, 3x_{1} + 4x_{2} \leq 24, x_{1} \geq 0, x_{2} \geq 0.



Solve then following LPP graphical method:

Maximize : Z = 7x + y

Subject to : 5x +y \geq 5, x + y \geq 3, x \geq 0, y \geq 0.



Solve the following LPP:
Maximize z = 5x_{1} + 6x_{2} subject to 2x_{1} + 3x_{2} \leq 18, 2x_{1} + x_{2} \leq 12, x_{1} \geq 0, x_{2} \geq 0.



Solve the following LPP by graphical method:
Mininize z = 8x + 10y,
Subject to 2x + y \geq 7, 2x + 3y \geq 15, y \geq 2, x \geq 0, y \geq 0.



Solve then following LPP graphical method:

Maximize : z = 6x + 21y

Subject to : x + 2y  \geq 3, x + 4y \geq 4, 3x + y \geq 3, x \geq 0, y \geq 0.



Solve the following problem graphically:
Minimize Z = -3x + 4y subject to x + 2y \leq 8; 3x + 2y \leq 12; x \geq 0, y \geq 0.



Solve the following problem graphically:
Maximize Z = 3x + 4y subject to the constraints : x + y \leq 4, x \geq 0, y \geq 0.



Show that the minimum of Z occurs at more than two points
Minimise and Maximise Z = x + 2y
subject to x + 2y \geq 100, 2x - y \leq 0, 2x + y \leq 200; x, y \geq 0.



Show that the minimum of Z occurs at more than two points
Maximise Z = x + y,
subject to x - y \leq -1, -x + y \leq 0, x, y \geq 0.



A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman's time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman's time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity ?
(ii) If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the maximum profit of the factory when it works at full capacity.



A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to, produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ? 17.50 per package on nuts and ? 7.00 per package on bolts. How many packages of each should be produced each day his machines for at the most 12 hours a day ? 



Solve the following  LPP:
Maximize  z = 60x + 50 y subject to
x + 2y \leq 40, 3x + 2y \leq 60, x \geq 0, y \geq 0.



Show that the minimum of Z occurs at more than two points
Maximise Z = -x + 2y, subject to the constraints:
x \geq 3, x + y \geq 5, x + 2y \geq 6, y \geq 0



Solve the following problem graphically:
Minimum Z = 3x + 5y such that x + 3y \leq 3, x + y \geq 2, x, y \geq 0.



One kind of cake requires 200\,g of flour and 25\,g of fat, and another kind of cake requires 100\,g of flour and 50\,g of fat. Find the maximum number of cakes which can be made from 5 \,kg of flour and 1 \,kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.



Minimize Z = - 50x + 20y
Subject to the constraints
2x - y \geq -5
3x + y \geq 3
2x- 3y \leq 12
and x \geq 0, y \geq 0



Minimize Z = -3x + Ay
subject to the constraints x + 2y \leq 8
3x + 2y \leq 12
and x \geq 0, y \geq 0



Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below : 
2x + 4y \leq 8  
3x + y \leq 6  
x + y \leq 4  
x \geq 0 , y \geq 0



One kind of cake required 300 \,g of flour and 15\,g of fat, another kind requires 150 \,gm of flour and 30 \,gm of fat. Find the maximum number of cakes which can be made from 7.5 \,kg of flour and 600\,gm of fat.



The marks obtained by a student of class XI in first and second terminal examination are 62 and 48, respectively. Find the minimum marks he should get in the annual examination to have an average of at least 60 marks.



Minimize Z = x + 2y
Subject to the constraints
2x + y \geq 3
x + 2y \geq 6
and x \geq 0,\,y \geq 0



Maximize Z = x + y
Subject to the constraints
x - y \leq -1
-x + y \leq 0
and x \leq 0,y \leq 0



Maximize
Z = -x + 2y
Subject to the constraints
x \geq 3
x + y \geq 5
x + 2y \geq 6 and
x \geq 0,y \geq 0



Two tailors, A and B, earn < 300 and < 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labor cost, formulate this as an LPP.



A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) if nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240kg of phosphoric acid at least 270kg of potash and at most 310 kg of chlorine.
If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?
kg per bag
Brand PBrand Q
Nitrogen33.5
Phosphoric acid12
Potash31.5
Chlorine1.52



There are two types of fertilizers {F}_{1} and {F}_{2}. {F}_{1} consists of 10% nitrogen and 6% phosphoric acid and {F}_{2} consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14kg of nitrogen and 14kg of phosphoric acid for her crop. If {F}_{1} cost Rs.6/kg and {F}_{2} costs Rs.5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?



A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin content of one kg of food is given below:
FoodVitamin AVitamin BVitamin C
X123
Y221
One kg of food X costs Rs.16 and one kg of food Y costs Rs.20. Find the least cost of the mixture which will produce the required diet?



A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) if nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240kg of phosphoric acid at least 270kg of potash and at most 310 kg of chlorine.
If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?
kg per bag
Brand PBrand Q
Nitrogen33.5
Phosphoric acid12
Potash31.5
Chlorine1.52



A merchant plans to sell two types of personal computers, a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant would stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.



A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below
Type of toysMachine
I
Machine
II
Machine
III
A12186
B609
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs.7.50 and that on each toy of type B is Rs.5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.



The graph of 3x + 2y = 6 meets the x-axis at point P and the y-axis at point Q. Use the graphical method,then the co-ordinates of points P and Q are P = (2, 0) and Q = (0, 3)
If true then enter 1 and if false then enter 0



An oil company has two depots A and B with capacities of 7000L  and 4000L  respectively. The company is to supply oil to three petrol pumps D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table:
Distance (in km)
From/ToAB
D73
E64
F32
Assuming that the transportation cost of 10 litres of oil is Re.1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?



A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of 80 on each piece of type A and 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?



You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximise storage volume. Write the maximum storage volume in the provided box.



A dietician wishes to mix two types of foods in such away that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin 1 unit/kg of vitamin C. It costs Rs. 5 per kg to purchase food I and Rs. 7 per kg to purchase Food II. Determine the maximum cost of such a mixture. Formulate the above as a LPP and solve it graphically.



A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs.\ 12 and Rs.\ 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?



A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?  



A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per heclare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit?
From an LPP from the above and solve it graphically. Do you agree with the massage that the protection of wildlife is utmost necessary to preserve the balance in environment?



A manufacturing company makes two types of teaching aids A and B of Mathematics for Class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs. 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of  type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?



A dietician wishes to mix two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contains of per kg food is given below.
Food    Vitamin A    Vitamin B     Vitamin C
X1 unit2 units3 units
Y2 units2 units1 unit
One kg of food X costs rupees 24 and one kg of food Y costs rupees 36. Using Linear Programming, find the least cost of the total mixture which will contain the required vitamins.



A dealer in rual area wishes to purchase a number of sewing machines. He has only Rs\ 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs\ 360 and a manually operated sewing machine Rs\ 240. He can sell an electronic sewing machine at a profit of Rs\ 22 and a manually operated sewing machine at a profit of Rs\ 18. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit? Formulate this problem mathematically and then solve it.



A company manufactures three kinds of calculators : A, B and C in its two factories I and II, The company has got an order for manufacturing at least 6400 calculators of kind A, 4000 of kind B and 4800 of kind C. The daily output of factory I is of 50 calculators of kind A, 50 calculators of kind B, and 30 calculators of kind C. The daily output of factory II is of 40 calculators of kind A, 20 of kind B and 40 of kind C. The cost per day to run factory I is Rs. 12,000 and of factory II is Rs. 15,000
How many days do the two factories have to be in operation to produce the order with the minimum cost? Formulate this problem as an LPP and solve it graphically.



A company manufactures two types of products A and B. Each unit of A requires 3 grams of nickel and 1 gram of chromium, while each unit of B requires 1 gram of nickel and 2 grams of chromium. The firm can produce 9 grams of nickel and 8 grams of chromium. The profit is Rs. 40 on each unit of product of type A and Rs. 50 on each unit of type B. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution



Find graphically, the maximum value of z=2x+5y, subject to constraints given below:
2x+4y\le 8
3x+y\le 6
x+y\le 4
x\ge 0, y \le 0.6



Minimize: Z=6x+4y
Subject to : 3x+2y \ge 12,
x+y\ge 5,
0\le x\le 5,
0\le y\le 6



A mill owner buys two types of machines A and B for his mill. Machine A occupies 1,000 sqm of area and requires 12 men to operate it; while machine B occupies 1,200 sqm of area and requires 8 men to operate it. The owner has 7,600 sqm of area available and 72 men to operate the machines. If machine A produces 50 units and machine B produces 40 units daily, how many machines of each type should he buy to maximize the daily output? Use Linear Programming to find the solution.



Solve the following L.P.P. graphically
Maximize: z = 10x + 25y
Subject to: x\leq 3 y\leq 3, x + y \leq 5, x \geq 0, y \geq 0



Two tailors P and Q eart Rs. 150 and Rs. 200 per day respectively. P can stitch 6 shirts and 4 trousers a day, while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce at least 60 shirts and 32 trousers at minimum labour cost?



A diet of a sick person must contains at least 48 units of vitamin A and 64 units of vitamin B. Two foods F_1 and F_2 are available. Food F_1 costs Rs. 6 per unit and Food F_2 costs Rs. 10 per unit. One unit of food F_1 contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F_2 contains 8 units of vitamin A and 12 units of vitamin B. Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimum nutritional requirements.



Solve the following linear programming problem graphically:
Maximize: Z=60x+40y
subject to the constraints:
x+2y\le 12; 2x+y\le 12
x+\cfrac{5}{4}y\ge 5;x\ge 0, y\ge 0.



A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toy in a day. The profit is Rs. 50 each on a toy of type A and Rs. 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution



Solve the following linear programming problem by graphical method:
Maximize Z = 3x + 2y
subject to the in constraints
x + 2y \leq 10, 3x + y \leq 15, x, y\geq 0



An aeroplane can carry a maximum load of 200 passengers. Baggage allowed to the first class ticket holder is 30 kg and for the economy class ticket holder is 20 kg. Maximum capacity of the aeroplane to carry the baggage is 4500 kg. The profit on each first class ticket is Rs.500 and on each economy class ticket is Rs.300. Formulate the problem, as L.P.P to maximize the profit.



Show that the minimum of Z occurs at more than two points.
Minimise and Maximise Z = x + 2y
subject to x + 2y \geq 100, 2x - y \leq 0, 2x + y \leq 200; x, y \geq 0.



Solve the following Linear Programming Problems graphically:
Minimise Z = x + 2y
subject to 2x + y \geq 3, x + 2y \geq 6, x, y \geq 0.



A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on hand-operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand-operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs. 7 and screws B at a profit of Rs. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.



Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8\ units of vitamin A and 11\ units of vitamin B. Food P costs Rs.\ 60/kg and Food Q costs Rs.\ 80/kg. Food P contains 3\ units/kg of Vitamin A and 5\ units / kg of Vitamin B while food Q contains 4\ units/kg of Vitamin A and 2\ units/kg of vitamin B. Determine the minimum cost of the mixture.



Show that the maximum of Z occurs at more than two points.
Minimise and Maximise Z = 5x + 10y subject to x + 2y \leq 120, x + y \geq 60, x - 2y \geq 60, x - 2y \geq 0, x, y\geq 0.



Show that the maximum of Z occurs at more than two points.
Maximise Z = -x + 2y, subject to the constraints:
x\geq 3, x + y \geq 5, x + 2y \geq 6, y \geq 0.



A manufacturing company makes two types of teaching aids A and B of Mathematics for Class X. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs. 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch.



A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs. 5 each for type A and Rs. 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?



Determine graphically the minimum value of the objective function.
Z = -50x + 20y
Subject to constraints
2x - y \geq -5
3x + y \geq 3
2x - 3y \leq 12
x \geq 0, y \geq 0.



Use the graphical method to solve each of the following LP problems.
A wheat and barley farmer has 168 hectare of ploughed land, and a capital of E2000. It costs E14 to sow hectare wheat ad E10 to sow one hectare of barley. 
Suppose that his profit is E80 per hectare of wheat and E55 per hectare of barley. Find the optimal number of hectares of wheat and barley that must be ploughed in order to maximise profit? What is the maximum profit?



Solve the following minimal assignment problem and hence find the minimum value:
IIIIIIIV
A21097
B132122
C3461
D41549



An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?



A merchant plans to sell two types of personal computers - a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.



Construct the graphs of the following functions.
\displaystyle y \, = \, (x \, + \, 3) \, / \, (x \, - \, 1).



Construct the graphs of the following functions. 
\displaystyle x \, = \, \sqrt{1 \, - \, | y |}.
The maximum possible value of x is:



A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other types by at most 600 units. If the company makes a profit of Rs. 12 and Rs. 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?



A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F_{1} and F_{2} are available. Food F_{1} costs Rs. 4 per unit food and F_{2} costs Rs. 6 per unit. One unit of food F_{1} contains 3 units of vitamin A and 4 units of minerals. One unit of food F_{2} contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimal nutritional requirements.



A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18\ units, 45\ units and 24\ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?



An company manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each of fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. 
Each air conditioner sold yields a profit of E25. Each fan assembled may be sold for a profit of E15. Formulate and solve this linear programming mix situation to find the best combination of air conditioners and fans that yields the highest profit.



Solve graphically:
x+2y\le 10;x+y\ge 1;x-y\le 0;\le 0



A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and a half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs\100 and that on a bracelet is Rs\ 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.



An advertising company wishes to plan its advertising strategy in three different media-television,radio and magazines.The purpose of advertising is to reach as large a number of potential customers as possible.Following data have been obtained from market survey:
StrategiesTelevisionRadioMagazine-IMagazine-II
Cost of an advertising companyRs.30,000Rs.20,000Rs.15,000Rs.10,000
No.of potential customers reached per unit2,00,0006,00,0001,50,0001,00,000
No.of female customers reached per unit1,50,0004,00,00070,00050,000
The company wants to spend not more than Rs.4,50,000 on advertising.Following are the further requirements that must be met:
\left(i\right)at least 1 million exposures take place among female customers.
\left(ii\right)advertising on magazines be limited to Rs.1,50,000
\left(iii\right)at least 3 advertising units be bought on magazine I and 2 units on magazine II
\left(iv\right)the number of advertising units on television and radio should each be between 5 and 10.
Formulate an L.P model for the problem.



Solve the following system of inequalities graphicallyx\ge 3, y\ge 2



David wants to invest at most Rs.\ 12,000 in bonds A and B. According to the rule, he has to invest at least Rs.\ 2,000 in Bond A and at least Rs.\ 4,000 in Bond B. If the rates of interest on Bonds A and B respectively are 8\% and 10\% perannum, formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest received in a year.



Find the number of integer solutions of {x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}=20 where {x}_{1}\ge -5, {x}_{2}\ge 3, {x}_{3}\ge 0, {x}_{4}\ge 1



The following table gives production yields per hutone whest of 100 farms of a village
Production Yield50-5555-6060-6565-7070-7575-80
Number of Farms2812243846

Change the distribution to a more then type distribution and draw its vgive.



To solve the following L.P.P.
Minimize z=30x+20y
Subject to the constraint, x+y\leq 8, x+2y \geq 4, 6x+7y\geq 12, x\geq 0; y\geq 0.



Solve the following L.P.P. graphically .Minimize Z=3x+5y .S
Subject to constraints
2x+3y \ge 12
-x+y\le3
x\le 4
y\ge 3



Minimize and maximize z=3x+9y subject to the constraints
x+3y\le 60
10\le x+y
x\le y
0\le x,y
by the graphical method. Minimum and maximum values are p and q then p+q  =



A manufacturer makes two types of toys A and B. Three machine are need for this purpose and the time (In minutes) required for each toy on the machines is given below
Types of toysIIIIII
A201010
B102030
The machine I,II and III are available for a machine of 3 hours, 2 hours and 2 hours 30 minutes respectively. The profit on each on each toy of type A is Rs. 50 and that of type B is Rs.\ 60 
Formulate the above problem as a LPP and solve it graphically to maximize profit.



Solve each of the following linear programming problem by graphical method.
Maximize Z=50x+30y
Subject to 2x+y\le 18
                 3x+2y\le 34
                   x,y\ge 0



A manufactures can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operation: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.
AB
Grinding 12
Turning 31
Assembling 63
Testing 54
The available capacities of these operations in hours for the given time period are: grinding 30; turning 60 assembling 200; testing 200. The contribution to profit is Rs. 2 for each unit of A and Rs. 3 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.



Solve each of the following linear programming problem by graphical method.
Minimize Z=5x+3y
Subject to 2x+y\ge 10
                 x+3y\ge 15
                   x\le 10
                   y\le 8
                  x, y\ge 0



A company produces two types of goods, A and B that require gold and silver. Each unit of type A requires 3 gm of silver and 1 gm of gold while that of type B requires 1 gm of silver and 2 gm of gold. The company can produce 9 gm of silver and 8 gm of gold. If each unit of type A brings a profit of Rs. 40 and that of type B Rs. 50, find the number of units of each type that the company should produce to maximize the profit. What is the maximum profit?



A furniture manufacturing company plans to make two products: chairs and tables. From its available resources which consists of 400 square feet of teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs. 45, while each table uses 20 square feet of word and 15 man-hours and yields a profit of Rs. 80. How many items of each product should be produced by the company so that the profit is maximum?



Two tailors, A and B earn Rs\ 15 and Rs\ 20 per day respectively. A can stitch 6 shirts and 4 pants while b can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?



A manufacturer makes two types A and B of tea-cups.Three machines are needed for the manufacturer and the time in  minutes required for each cup on the machines is given below:
Machines
IIIIII
A12186
B609
Each machine is available for a maximum of 6 hours per day. If the profit on each cup A is 75 paise and that on each cup B is 50 paise, show that 15 tea-cups of type A and 30 of type B should be manufacturer in a day to get the maximum profit.



A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given as at least 240 kg of phosphoric acid,  270 kg of potash and at most 310 kg of chlorine.
kg per bag
Brand PBrand Q
Nitrogen33.5
Phosphoric acid 12
Potash 31.5
Chlorine1.52
If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?



A manufacturer of Furniture make two products : chairs and tables. Processing of these products is done on two machine A and B. A chair requires 2 hr on machine A and 6 hrs on machine B. A table requires 4 hrs on machines A and 2 hrs on machine B. There are 16 hrs of time per day available on machine A and 30 hrs on machine B. Profit gained by the manufacturer from a chair and a table is Rs. 3 and Rs. 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.



A company produces two types of leather belts, say type A and B. Belt A is a superior quality and belt B is of a lower quality. Profits on each type of belt are Rs. 2 and Rs. 1.50 per belt, respectively. Each belt of type B, the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 fancy buckles are available for this per day. For belt of type B, only 700 buckles are available per day. How should the company manufacture the two types of belts in order to have a maximum overall profit?



If a young man drives his scooter at a speed of 25 km/hr, he to spend Rs. 2\ \text{per km} on petrol. If he drives the scooter at a speed of 40 km/hour, it produces air pollution and increases his expenditure on petrol to Rs. 5\ \text{per km}. He has a maximum of Rs. 100 to spend on petrol and travel a maximum distance in one hour time with less polution. Express this problem as an LPP and solve it graphically. What value do you find here? 



A factory owner purchase  two types of machines, A and B for his factory. The requirements and limitations for the machines are as follows:
Area occupied by the machineLabour force for each machineDaily output in units
Machine A1000.sq.\ m12\ men60
Machine B1200\ sq.\ m8\ men40
He has an area of 7600 sq.m available and 72 skilled men who can operate the machines. How many machines of each type should he buy to maximize the daily output?



A factory uses three different resources for the manufacture of two different products 20 units of the resources A, 12 units of B and 16 units of C being available. 1 unit of the first produce requires 2,2 and 4 units of the respective resources and 1 unit of the second product requires 4,2 and 0 units of respective resources. It is known that the first product gives a profit of 2 monetary units per unit and the second 3. Formulate the linear programming problem. How many units of each product should be manufactured for maximizing the profit? Solve it graphically.



A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs. 1400 to purchase young trees. He has the choice of two years. Type A requires 10 sq.m of ground per tree and costs Rs. 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs. 25 per tree. When fully grown type A produces an average of 20 kg of fruit which can be sold at a profit of Rs. 2.00 per kg and type B produces an average 40 kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when when the trees are fully grown? What is the maximum profit?



A producer has 30 and 17 units of labour and capital respectively which he can use to produce two type of goods X and Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Y are priced at Rs. 100 and Rs. 120 per unit respectively, how should be producer use his resources to maximize the total revenue? Solve the problem graphically.



A manufacture consider that men and women worker are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produces two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resource to maximize the total revenue? From the above as an LPP and solve graphically. Do you agree with this total view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?



A furniture dealer deals in only two items: tables and chairs. He has RS 30000 to invest and a space to store at most 60 pieces. A table costs him RS 1500 and a chair RS 300. Formulate the data in the of inequatins and draw a graph representing the solution of these inequations.



A manufacturer makes two products A and B. Product A sells at Rs. 200 each and takes \dfrac {1}{2} hour to make. Product B sells at Rs. 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hours of production and weekly turnover must not be less than Rs. 10000. If the profit on each of product A is Rs. 20 and on product B is Rs. 30, then how many of each should be produced so that the profits is maximum. Also, find the maximum profit.



A firm manufactures two products A and B. Each product is processed on two machines M_{1} and M_{2}. Product A requires 4 minutes of processing time on M_{1} and 8 min. on M_{2}; product B requires 4 minutes on M_{1} and 4 min. on M_{2}. The machine M_{1} is available for not more than 8 hrs 20 min. while machine M_{2} is available for 10 hrs during any working day. The products A and B are sold at a profit of Rs. 3 and Rs. 4 respectively.
Formulate the problem as a linear programming problem and find how many product of each type should be produced by the firm each day in order to get maximum profit.



A firm manufacturing two types of electric items, A and B can make a profit of Rs. 20 per unit of A and Rs. 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programming problem for maximum profit and solve ir graphically.



A chemical company produces two compounds A and B. The following table gives the units of ingredients, C and D per kg of compounds A and B as well as minimum requirement of C and D and costs per kg of A and B. Find the quantities of A and B which would give a supply of C and D at a minimum cost.
CompoundsMinimum requirement
AB
Ingredient C1280
Ingredient D3175
Cost (in Rs) per\ kg46



Two godowns, A and B grain storage capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirement are 60,50 and 40 quintals respectively. The cost transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in Rs)
To \downarrow \ From \toAB
D6.004.00
E3.002.00
F2.503.00
How should the supplies be transported in order that the transportation cost is minimum?



To receive grade A in a course one must obtain an average of 90 marks or more in five papers, each of 100 marks. If Tanvy scored 89, 93, 95 and 91 marks in first four papers, find the minimum marks that she must score in the last paper to get grade A in the course.



Class 12 Commerce Maths Extra Questions