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CBSE Questions for Class 10 Maths Some Applications Of Trigonometry Quiz 5 - MCQExams.com
CBSE
Class 10 Maths
Some Applications Of Trigonometry
Quiz 5
The angle of elevation of the top of a lamp-post as observed from a point
40
m
distant from the foot of the post, is
30
0
.
The height of the lamp post is
Report Question
0%
40
√
3
m
0%
40
√
3
3
m
0%
20
m
0%
40
√
3
2
Explanation
Consider the given triangle
Let AB be the height of the lamp post denoted by h
Let BC bet the horizontal distance of the point from the foot of the lamp post
Given that angle
C
=
30
0
Therefore
tan
30
0
=
1
√
3
=
A
B
B
C
=
h
40
h
=
40
√
3
Rationalising the denominator we get
h
=
40
√
3
3
Hence answer is B
A light-house
100
m high observe that two ships are approaching it from West and South respectively. If the angles of depression of the two ships are
30
∘
and
45
∘
respectively, then the distance between the two ships is -
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0%
100
(
√
3
+
1
)
m
0%
100
(
√
3
+
1
)
2
m
0%
200
m
0%
400
m
Explanation
we can observe that in
△
A
B
C
tan
45
o
=
100
a
⇒
a
=
100
Similarly, in
△
A
C
D
tan
30
o
=
100
b
⇒
b
=
100
√
3
So, in
△
B
C
D
(Top View)
a
2
+
b
2
=
d
2
⇒
d
2
=
10000
+
30000
=
40000
⇒
d
=
200
An electric pole
10
m high is tied to three steel wires which are fixed to the ground at points which are at equal distances from the foot of the pole. These points, if joined form an equilateral triangle. If the steel wires are each inclined to the ground at
60
∘
, then each side of the equilateral triangle measures
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0%
10
m
0%
10
√
3
m
0%
10
√
3
m
0%
None of the above
Explanation
Let C, D, F be the points at which are steel wires are fixed to the ground. B is the foot of the pole.
From fig. (ii),
cot
60
∘
=
C
B
A
B
∴
or
CB = \cfrac{10}{\sqrt{3}}
From fig. (i),
\cfrac{CE}{CB} = \cos 30^{\circ}
\therefore CE = \cfrac{CB \times \sqrt{3}}{2} = \cfrac{10}{2} = 5, CD = 10
Calculate the height AD.
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0%
12.12
0%
8.96
0%
5.14
0%
18.24
Explanation
tan 75^{\circ} = \frac{AD}{2.4}
tan 75^{\circ} = tan (45^{\circ} + 30^{\circ})
\displaystyle =\frac{tan 45^{\circ} + tan 30^{\circ}}{1 - tan 45^{\circ} tan 30^{\circ}} = \frac{1+\frac{1}{\sqrt{3}}}{1- \frac{1}{\sqrt{3}}} = \frac{\sqrt{3}+1}{\sqrt{3}-1}
=\displaystyle \frac{(\sqrt{3} + 1)^2}{3-1} = \frac{1+3 + 2 \sqrt{3}}{2} = 2 + \sqrt{3} = 3.732
AD = 2.4 \times tan 75^{\circ} = 8.96
A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the posts is a right angle. If the height of the two posts are two times and four times the height of the man and the distance between the poles is equal to the height of the largest pole, then the ratio of the distance of the man from the shorter and the longer post is
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0%
3 :1
0%
2 : 3
0%
3 : 2
0%
1 : 3
Explanation
Let
AB
be the men of height
h\ m
,
CD
and
EF
be the two towers of height
2h\ m
and
4h\ m
respectively.
\therefore \angle CAE={ 90 }^{ o }
Draw
PQ\parallel DF
passing through
A
.
\Rightarrow CP=PD=h\ m,QF=h\ m
\therefore EQ=3h\ m
Let
DB
be
d\ m
\therefore DB=PA=d\ m
and
BF=AQ=\left( 4h-d \right) m
PAQ is a straight line.
\therefore \angle PAC+\angle CAE+\angle EAQ={ 180 }^{ o }
\angle PAC+{ 90 }^{ o }+\angle EAQ={ 180 }^{ o }
\therefore \angle PAC+\angle EAQ={ 90 }^{ o }
Let
\angle PAC
be
x
.
\therefore \angle EAQ={ 90 }^{ o }-x
...(1)
In
\triangle CPA,\angle PAC=x
\Rightarrow \angle CPA={ 90 }^{ o }-x
...(2)
In
\triangle CPA
and
\triangle AQE
\angle PCE=\angle EAQ
\left[ each{ 90 }^{ o }-x \right]
\angle CPA=\angle EAQ
\left[ each{ 90 }^{ o } \right]
\therefore \triangle CPA\sim \triangle AQE
\displaystyle\Rightarrow \frac { CP }{ AQ } =\frac { PA }{ QE } =\frac { CA }{ AE }
(ESST)
\displaystyle\therefore \frac { h }{ 4h-d } =\frac { d }{ 3h }
\Rightarrow 3{ h }^{ 2 }=d\left( 4h-d \right)
\Rightarrow 3{ h }^{ 2 }=4dh-{ d }^{ 2 }
\left( d-3h \right) \left( d-h \right) =0
d-3h=0
or
d=h
when
d=3h
\displaystyle\frac { DB }{ BF } =\frac { 3h }{ h } =\frac { 3 }{ 1 } =3:1
when
d=h
\displaystyle\frac { DB }{ BF } =\frac { h }{ 3h } =\frac { 1 }{ 3 } =1:3
Henc,e option
D
is correct.
A vertical lamp-post of height 9 meters stands at the corner of a rectangular field.The angle of elevation of its top from the farthest corner is
\displaystyle 30^{\circ}
,while from another corner it is
\displaystyle 45^{\circ}.
The area of the field is
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0%
\displaystyle 9\sqrt{2}\: meter\:^{2}
0%
\displaystyle 81\sqrt{3} \:meter\:^{2}
0%
\displaystyle 81\sqrt{2}\: meter\:^{2}
0%
\displaystyle 9\sqrt{3}\: meter\:^{2}
Explanation
Consider rectangle
ABCD
be the rectangular field. Let
PD
be the height of vertical lamp post.
it is given that Height of the Pole
=PD=9
m
Now in
\triangle PDA
tan45^o=\dfrac{PD}{DA}=\dfrac{9}{DA}\Rightarrow DA=9
Now in
\triangle PDB
tan30^o=\dfrac{9}{DB}\Rightarrow DB=9\sqrt3
Also, we can see that
\triangle DAB
is Right Angled Triangle
\therefore DA^2+AB^2=DB^2\Rightarrow 9^2+AB^2=(9\sqrt3)^2\Rightarrow AB^2=81\times2\Rightarrow AB=9\sqrt2
Now, Area of Rectangle
=DA\times AB=9\sqrt2\times 9=81\sqrt2
m^2
Therefore, option (C) is the correct answer.
When a eucalyptus tree is broken by strong wind, its top strikes the ground at an angle of
30^{\circ}
and at a distance of
15
m from the foot. What is the height of the tree?
Report Question
0%
15
\sqrt{3}
0%
10
\sqrt{3}
m
0%
20
m
0%
10
m
Explanation
Let the height of the tree be
A'C.
'h'
is the height of the tree from the broken point to the bottom.
Now, In
\triangle BAC,
\Rightarrow \tan { 30° } =\dfrac { AC }{ BC } =\dfrac { h }{ 15 }
\Rightarrow \dfrac { 1 }{ \sqrt { 3 } } =\dfrac { h }{ 15 }
\Rightarrow h=\dfrac { 15 }{ \sqrt { 3 } } m
Again,
\cos { 30° } =\dfrac { BC }{ BA } =\dfrac { 15 }{ BA }
\Rightarrow \dfrac { \sqrt { 3 } }{ 2 } =\dfrac { 15 }{ BA }
\Rightarrow BA=\dfrac { 30 }{ \sqrt { 3 } } m
Now, the height of tree
=AB+AC
=\dfrac { 30 }{ \sqrt { 3 } } +\dfrac { 15 }{ \sqrt { 3 } }
=\dfrac { 45 }{ \sqrt { 3 } }
=\dfrac { 45\sqrt { 3 } }{ 3 }
=15\sqrt { 3 } m
Hence, the answer is
15\sqrt { 3 } m.
STATEMENT -
1
: The angle of elevation of a point viewed is the angle formed by the line of sight with the horizontal when the point being view is above the horizontal level.
STATEMENT -
2
: Then the angle of depression of a point view is the angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.
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0%
Statement -
1
is True, Statement -
2
is True, Statement -
2
is a correct explanation for Statement -
1
0%
Statement -
1
is True, Statement -
2
is True : Statement
2
is NOT a correct explanation for statement -
1
0%
Statement -
1
is True, Statement -
2
is False
0%
Statement -
1
is False, Statement -
2
is True
Explanation
Both the statements are true but the second statement is not a correct explanation of the first statement.
Hence, the answer is option
B.
A person, standing on the bank of a river, observes that the angle subtended by a tree on the opposite bank is
60^{\circ}
when he retreats
20
m from the bank, he finds the angle to be
30^{\circ}
. The height of the tree and the breadth of the river.
Report Question
0%
10
\sqrt{3}
m,
10
m
0%
10
;
10
\sqrt{3}
m
0%
20
m,
30
m
0%
None of these
Explanation
\textbf{Step-1: Draw the appropriate diagram & label it.}
\text{Let AB be the breadth of the river and BC be the height of the}
\text{tree which makes a}
\angle
\text{of 60}
^{\circ}
\text{at a point A on the opposite bank.}
\text{Let D be the position of the person after retreating 20 m from the bank.}
\text{Let AB}
= x
\text{metres and BC}
= h
\text{metres.}
\text{From right angled}
\triangle
\text{ABC and DBC,}
\text{we have,}
\tan 60^{\circ} = \cfrac{BC}{AB}
\text{and}
\tan 30^{\circ} = \cfrac{h}{20+x}
\Rightarrow \sqrt{3} = \cfrac{h}{x}
\text{and}
\cfrac{1}{\sqrt{3}} = \cfrac{h}{x+20}
\textbf{Step-2: Use the above results & get the required unknown.}
\Rightarrow h = x \sqrt{3}
\text{and}
h = \cfrac{x+20}{\sqrt{3}}
\Rightarrow x \sqrt{3} = \cfrac{x+20}{\sqrt{3}} \Rightarrow 3x = x+20 \Rightarrow x=10 m
\text{Putting}
x=10
\text{in}
h = \sqrt{3}x
,
\text{we get,}
h= 10 \sqrt{3} = 17.32 m
\textbf{Hence, the height of the tree = 17.32 m and the breadth of the river =10 m.}
A tree is broken by the wind. The top struck the ground at an angle of
30^{\circ}
and at a distance of 30 m from the root, then the whole height of the tree.
(\sqrt3 =1.73)
is 51.9 m
If true then enter
1
and if false then enter
0
Report Question
0%
TRUE
0%
FALSE
0%
can't say
0%
NONE
Explanation
Angle made by the broken part with ground =
30^{\circ}
broken tree is at distance (d) = 30 m from the roots
Let broken part be of length = b
and unbroken part of length = u
Now, the broken part, ground and the unbroken part of the tree forms a right angled triangle
Thus,
\tan 30 = \dfrac{u}{d}
u = \dfrac{30}{\sqrt{3}}
u = 10\sqrt{3}
\sin 30 = \dfrac{u}{b}
$
\dfrac{1}{2} = \dfrac{10\sqrt{3}}{b}
b = 20\sqrt{3}
Thus, length of tree =
u + b
=
10\sqrt{3} + 20\sqrt{3} = 30\sqrt{3}= 51.96
m
A kite is attached to a
100\ m
long string. Find the greatest height reached by the kite when its string makes an angle of
\displaystyle 60^{\circ}
with the level ground.
Report Question
0%
86.6
0%
45.7
0%
63.8
0%
72.0
Explanation
The highest height the kite reaches be P
The length of string is L = 100 m
The angle made by the string with the ground =
60^{\circ}
now,
\sin 60 = \dfrac{P}{L}
P = \dfrac{100 \sqrt{3}}{2}
P = 50 \sqrt{3}
P = 86.6
m
Choose the correct option for the following.
A kite is flying at a height of
60
m above the ground. The string attached to the kite is temporarily tied to the ground. The inclination of the string with the ground is
60^{\circ}
, then the length of the string, assuming that there is no slack in the string is
69.2
m.
(\sqrt3 = 1.73)
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0%
True
0%
False
0%
Ambiguous
0%
Data insufficient
Explanation
Height of the kite =
60
m
Angle made by the string with the ground, =
60^{\circ}
.
The kite, string, and the ground form a right-angled triangle
Thus,
\sin 60 = \dfrac{60}{string}
\dfrac{\sqrt{3}}{2} = \dfrac{60}{string}
string= \dfrac{120}{\sqrt{3}}
Length of string
= 69.2
m
Choose the correct option for the following.
From the top of a lighthouse, an observer looks at a ship and finds the angle of depression to be
60^{\circ}
. If the height of the lighthouse is 90 meters then that ship is 51.9 m from the lighthouse.
\displaystyle (\sqrt3 = 1.73)
.
Report Question
0%
True
0%
False
0%
Ambiguous
0%
Data insufficient
Explanation
Given, height of lighthouse =
90
m
Angle of depression =
60^{\circ}
The angle of elevation from the ship to the top of lighthouse = angle of depression =
60^{\circ}
Now,
\tan 60 = \dfrac{height}{distance}
\sqrt{3} = \dfrac{90}{distance}
distance = 51.9
m
Two buildings are in front of each other on either side of a road of width
10
metres. From the top of the first building, which is
30
metres high, the angle of elevation of the top of the second is
45^{\circ}
. What is the height of the second building (in metres)?
Report Question
0%
10
0%
30
0%
40
0%
25
Explanation
CD=EF=10
we can see that
CDFE
is a rectangle,
\therefore CE=DF=30\\ \& \ CD=EF=10
Now, in
\triangle ACD
\tan45^o=\dfrac{AC}{10}\Rightarrow AC=10
Therefore, height of the second Building
=AC+CE=10+30=40
Therefore, Answer is
40
A ladder is placed against a vertical tower. If the ladder makes an angle of
\displaystyle 30^{\circ}
with the ground and reaches up to a height of
15\ m
of the tower; find length of the ladder in cm.
Report Question
0%
2311
0%
3000
0%
1688
0%
1200
Explanation
Consider the ladder and wall system as the right angled triangle, such that length of wall (W) till the ladder reaches be the perpendicular= 15 cm and angle (G) made by ladder (L) with the ground is
30^{\circ}
,
and length of ladder be the Hypotenuse.
Thus,
\sin G = \dfrac{P}{H}
\sin 30 = \dfrac{W}{L}
\dfrac{1}{2} = \dfrac{15}{L}
L = 30\ m=3000\ cm
Two poles of height 18 metres and 7 metres are erected on the ground. A wire of length
22
metres tied to the top of the poles. Find the angle made by the wire with the horizontal.(in degree)
Report Question
0%
30
0%
10
0%
20
0%
15
Explanation
Angle of elevation from smaller pole to bigger =
x
Height of smaller building
= 7
m
Height of bigger pole
= 18
m
Length of the rope
= 22
m
Difference between the two buildings
= 11
m
Now,
\sin x = \cfrac{opposite\ side}{hypotaneous}
\sin x = \cfrac{11}{22} = \dfrac{1}{2}
\sin x = \sin 30
x = 30^{\circ}
Determine
x
.
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0%
x=4m
0%
x=5m
0%
x=6m
0%
x=7m
Explanation
Let
AB
be the tree broken by the wind at the point
C
. Its top
B
strike the ground at the point
D
such that
\angle CDA=\theta
and
CB
takes the position of
CD
i.e.,
CD=CB=y
metres. But
AC=x
metres then the height of the tree
=AB=(x+y)=15m, CD=y=(15-x)
metres
ADC
is rt. angles
\triangle
at
A
{CD}^{2}={AC}^{2}+{AD}^{2}
\Rightarrow
{(15-x)}^{2}={x}^{2}={(5\sqrt 3)}^{2}={x}^{2}+75
(Pythagoras theorem)
\Rightarrow
225+{x}^{2}-30x={x}^{2}+75
\Rightarrow
30x=150m
\Rightarrow
x=5m
\therefore
The tree is broken at a height of
5m
from the bottom or ground.
A man
1.8m
tall stands at distance of
3.6m
from a lamp post and casts a shadow of
5.4m
on the ground. Find the height of the lamp post.
Report Question
0%
2m
0%
5m
0%
3m
0%
7m
Explanation
Let
\theta
be the unknown angle of elevation.
Let DF be the man and AB be the lamp post.
Considering the given figure.
\tan\theta
=\dfrac{1.8}{5.4}
=\dfrac{x}{3.6}
Hence
x=\dfrac{(3.6).(1.8)}{5.4}
=1.2m
Hence
The height of the lamp post is
1.2m+1.8m
=3m
A lamp post
\displaystyle 5\sqrt{3}
m high casts a shadow 5 long on the ground. The Sun's elevation at this point is
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0%
\displaystyle 30^{\circ}
0%
\displaystyle 45^{\circ}
0%
\displaystyle 60^{\circ}
0%
\displaystyle 90^{\circ}
Explanation
Given,
Height of the post is
5\sqrt 3
and base is
5
m
\tan θ=\dfrac{AB}{BC}
\therefore \tan θ=\dfrac{5\sqrt 3}{5}
=\sqrt 3
∴
\therefore θ=60^{0}
............ [since
\tan60^{0}=\sqrt 3
]
find the height of the tower from the cliff.
Report Question
0%
30
metres
0%
60
metres
0%
80
metres
0%
40
metres
Explanation
Let
PM
be the height of the cliff whose height is
60
metres.
P
be its top such that
PM=
height of cliff
=60m
. Let
BA
be the tower. Through
P
draw
PC
horizontally then by the conditions of th e questions.
\angle CPA={30}^{o}
and
\angle CPB={60}^{o}
Let the required height of the tower
AB
or
BA
=h
metres.
From
A
draw
AD\bot PM
, then:
\angle CPA=alt. \angle DAP
\Rightarrow
\angle DAP={30}^{o}
\angle CPB=alt. \angle MBP
\Rightarrow
\angle MBP={60}^{o}
Now
AD=BM....(1)
[Note this step carefully]
DP=MP-MD=MP-BA=(60-h)
metres
From the right
\triangle ADP
,
\cfrac{AD}{DP}=\cot {{30}^{o}}=\sqrt 3
AD=DP.\sqrt 3
AD=(60-h)\sqrt 3........(2)
And from the right
\triangle BMP
\cfrac{BM}{MP}=\cot {{60}^{o}}=\cfrac{1}{\sqrt 3}
BM=\cfrac{MP}{\sqrt 3}=\cfrac{60}{\sqrt 3}
BM=20\sqrt 3...........(3)
Using
(2)
and
(3)
in
(1)
, we get:
AD=BM
\Rightarrow
(60-h)\sqrt 3=20\sqrt 3
\Rightarrow
h=40m
Two poles of equal heights are standing opposite to each other on either side of a road, which is
80
metres wide. From a point between them on the road, the angles of elevation of their top are
{30}^{o}
and
{60}^{o}
. Find the position of the point and also the height of the poles.
Report Question
0%
50m
0%
35m
0%
20m
0%
10m
Explanation
Let
AB
and
CD
be two poles of equal height standing opposite to each other on either side of the road
BD
\Rightarrow
AB=CD=h
metres.
Let
P
be the observation point on the road
BD
. The angles of elevation of their top are
{30}^{o}
and
{60}^{o}
.
\angle APB={30}^{o}
,
\angle CPD={60}^{o}
The width of the road
=BD=80m
, let
PD=x
metres
Then
BP=(80-x)
metres consider right
\triangle CDP
, we have:
\cfrac{CD}{PD}=\tan{{60}^{o}}
\Rightarrow
\cfrac{h}{x}=\sqrt 3 \Rightarrow h=\sqrt 3x.........(1)
In right
\triangle ABP
we ave:
\cfrac{AB}{BP}=\tan{{30}^{o}} \Rightarrow \cfrac{h}{80-x}=\cfrac{1}{\sqrt 3} \Rightarrow h=\cfrac{80-x}{\sqrt 3}......(2)
From
(1)
and
(2)
we get
h=\sqrt 3x
h=\cfrac{80-x}{\sqrt 3}
\Rightarrow
(80-x)=3x \Rightarrow 4x=80 \Rightarrow x=20
Height of each pole
=AB=CD=\sqrt 3.x=20\sqrt 3=20(1.732)=34.64
metres
Position of point
P
is
20
from the first and
60m
from the second pole.
i.e., the position of the point
P
is
20m
from either of the poles.
the height of the post.
Report Question
0%
5\sqrt2m
0%
15m
0%
7\sqrt3m
0%
19m
Explanation
Let
AB
be the vertical post its shadow is
21m
when the altitude of the sun is
{30}^{o}
BC=21m,\angle ACB={30}^{o},AB=h
metres (figure)
ABC
is rt.
\triangle,\cfrac{AB}{BC}=\tan{{30}^{o}}
\Rightarrow
\cfrac{h}{21}=\cfrac{1}{\sqrt 3}
\Rightarrow
h=\cfrac{21}{\sqrt 3}=\cfrac{7\times \sqrt 3\times \sqrt 3}{\sqrt 3}=7\sqrt 3m
\Rightarrow
AB=h
, Height of the pole
=7\sqrt 3m
A boy
1.7
m tall, is
25
m away from a tower and observes the angle of elevation of the top of the tower to be
{60}^{o}
. Find the height of the tower.
Report Question
0%
55
m
0%
45
m
0%
20
m
0%
35
m
Explanation
Let
AA'
be the boy and the tower be
B'C
Therefore,
AA'=1.7
m
=BB'
and
A'B'=25\ m=AB
Now, angle of elevation is
\angle CAB=\theta
.
Given,
\theta=60^o
\Rightarrow \tan\theta=\dfrac{BC}{AB}
\Rightarrow 25\tan 60^o=BC
\Rightarrow BC=25\sqrt3
Now, height of tower
=BC+BB'
=25\sqrt3+1.7
=45
m
A captain of an aeroplane flying at an altitude of
1000
metres sights two ships as shown in the figure. If the angle of depression are
{60}^{o}
and
{30}^{o}
. Find the distance between the ships.
Report Question
0%
2210
m
0%
2196
m
0%
2219.7
m
0%
2309.3
m
Explanation
Let
A
be the position of the captain of an aeroplane flying at the altitude of
1000
metres from the ground.
AB=
the altitude of the aeroplane from the ground
=1000
m
P
and
Q
be the position of two ships.
Let
PB=x
metres and
BQ=y
metres
Required:
PQ=
Distance between the ships
=(x+y)
metres
ABP
is right angled
\triangle
at
B
\cfrac{AB}{PB}=\tan {{60}^{o}}
\cfrac{1000}{x}=\sqrt 3 \Rightarrow x=\cfrac{1000}{\sqrt 3}
x=\cfrac{1000(1.732)}{3}=577.3
m
ABQ
is right angled
\triangle
at
B
\cfrac{AB}{BQ}=\tan {{30}^{o}}
\cfrac{1000}{y}=\cfrac{1}{\sqrt 3} \Rightarrow y=1000\sqrt 3
y=1000(1.732)=1732
m
Required distance between the ships
=(x+y)
metres
=(577.3+1732)
m
=2309.3
m
The length of a string between a kite and a point on the roof of a building
10m
high is
180m
. If the string makes an angle
\theta
with the level ground such that
\tan {\theta}=\dfrac43
, how high is the kite from the ground?
Report Question
0%
154\ m
0%
176\ m
0%
198\ m
0%
214\ m
Explanation
\tan\theta=\dfrac{4}{3}
Hence
\sin\theta=\dfrac{4}{5}
and
\cos\theta=\dfrac{3}{5}
Hence the perpendicular height of the kite from the rooftop of the building is
=180\sin\theta
=180.\dfrac{4}{5}
=36(4)=144m
Hence the height of the kite from the ground will be
=
Height of the building
+
height of the kite from the rooftop of the building.
=144+10
=154\ m
The Shadow of a tower when the angle of elevation of the sun is
\displaystyle 30^{\circ}
is found to be
5 m
longer than when it was
\displaystyle 45^{\circ}
then the height of tower in meter is
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0%
\displaystyle \frac{5}{\sqrt{3}+1}
0%
\displaystyle \frac{5}{2}(\sqrt{3}-1)
0%
\displaystyle \frac{5}{2}(\sqrt{3}+1)
0%
None of these
Explanation
Let the height of the tower be
h
\displaystyle \therefore
In
\displaystyle \Delta ABC
\displaystyle \tan 45^{\circ}=\frac{h}{x}
\displaystyle \therefore x=h
In
\displaystyle \Delta ABD
\displaystyle \tan 30^{\circ}=\frac{h}{x+5}
\therefore \displaystyle \frac{1}{\sqrt{3}}=\frac{h}{x+5}
\therefore h + 5 = \displaystyle h\sqrt{3}
\therefore \displaystyle h\left ( \sqrt{3}-1 \right )=5
\therefore h = \displaystyle \frac{5\left ( \sqrt{3}+1 \right )}{\sqrt{3}-1\left ( \sqrt{3}-1 \right )}=\frac{5}{2}\left ( \sqrt{3}+1 \right )m
The angle of elevation of a cloud from a point
200
metres above a lake is
{30}^{o}
and the angle of depression of the reflection of the cloud in the lake is
{60}^{o}
. Find the height of the cloud.
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0%
The height of the cloud
=400m
0%
The height of the cloud
=280m
0%
The height of the cloud
=340m
0%
None of these
Explanation
Let
QR
be the surface of the lake and
P
be the point of observation which is
200m
above
Q
i.e.,
PQ=200m
. Let
C
be the position of the cloud.
Through
C
draw
CR
perpendicular to th esurface of the lake and
D
the reflection of the cloud in the lake then
CR=RD=h
metres. (Assume) (laws of relection)
Through
P
draw
PQ\bot
on the surface of lake and
PM\bot CD
.
\angle CPM={30}^{o},\angle MPD={60}^{o}
MR=PQ=200m
, Let
PM=x
metres
CM=CR-MR=(h-200m)
, (
\because CR=h
metres)
DM=DR+RM=(h+200m)
, (
\because DR=h
metres)
In rt.
\triangle CMP
\cfrac{CM}{PM}=\tan {{30}^{o}}
\cfrac{h-200}{x}=\cfrac{1}{\sqrt 3}
x=\sqrt 3(h-200)........(1)
From rt.
\triangle PMD
,
\cfrac{MD}{PM}=\tan {{60}^{o}}
\cfrac{h+200}{x}=\sqrt 3
x=\cfrac{h+200}{\sqrt 3}........(2)
From
(1)
and
(2)
, we get
\sqrt (h-200)=\cfrac{h+200}{\sqrt 3}
\Rightarrow
3(h-200)=h+200
\Rightarrow
3h-h=600+200=800
\Rightarrow
2h=800
\Rightarrow
h=\cfrac{800}{2}=400
Required height of the cloud
=h=CR=400m
What is the angle of elevation of a vertical flagstaff of height
100\sqrt 3m
from a point
100m
from its foot.
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0%
{18}^{o}
0%
{30}^{o}
0%
{48}^{o}
0%
{60}^{o}
Explanation
Height of flagstaff (H) =
100 \sqrt{3}
Distance between the point and the flagstaff (D) = 100 m
Let the angle of elevation be
\theta
Then,
\tan \theta = \dfrac{D}{H}
\tan \theta = \dfrac{100\sqrt{3}}{100}
\tan \theta = \sqrt{3}
\tan \theta = \tan 60^{\circ}
\theta = 60^{\circ}
Upper part of a vertical tree which is broken over by the winds just touches the ground and makes as angle of
\displaystyle 30^{\circ}
with the ground if the length of the broken part is 20 metres then remaining part of the tree is of length
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0%
20
meters
0%
\displaystyle 10\sqrt{3}
metres
0%
10
metres
0%
\displaystyle 10\sqrt{2}
metres
Explanation
in
\triangle ABC
\sin30^o=\dfrac{x}{20}\Rightarrow x=10 \ meters
The angle of elevation of the top of a tower as seen from two points
A
&
B
situated the same line and at distance '
p
' and '
q
' respectively from the foot of the tower are complementary, then height of the tower is
Report Question
0%
pq
0%
\displaystyle \frac{p}{q}
0%
\displaystyle \sqrt{pq}
0%
none of these
Explanation
Let
OT
be the tower of height
h
Let
O
be the base of the tower
Let
A
and
B
be two points on the through the base such that
OA = p
and
OB = q
In
\displaystyle \Delta AOT
\displaystyle \tan \alpha =\frac{OT}{OA}=\frac{h}{p}
.....(i)
In
\displaystyle \Delta BOT
\displaystyle \Rightarrow \tan \left ( 90-\alpha \right )=\frac{OT}{OB}=\frac{h}{q}
or
\cot \alpha =\dfrac{h}{q}
......(ii)
Multiplying (i) and (ii), we have
\displaystyle \Rightarrow \tan \alpha \cot \alpha =\frac{h}{p}\times \frac{h}{q}
\displaystyle \Rightarrow \tan \alpha* \frac{1}{\tan \alpha }=\frac{h^{2}}{pq}
\displaystyle \Rightarrow 1=\frac{h^{2}}{pq}
\displaystyle \Rightarrow h^{2}=pq
\displaystyle \Rightarrow h=\sqrt{pq}
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