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CBSE Questions for Class 10 Maths Some Applications Of Trigonometry Quiz 6 - MCQExams.com
CBSE
Class 10 Maths
Some Applications Of Trigonometry
Quiz 6
The length of a string between a kite and a point on the ground is 90m. The string makes an angle of
60
∘
with the level ground if there is no slack in the string, the height of the kite is
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0%
90
√
3
m
0%
45
√
3
m
0%
180
m
0%
45
m
Explanation
Let
A
be the kite and
A
C
is
90
m
In
ΔABC
\sin 60^{0}=\dfrac{AB}{AC}
=\dfrac{h}{90}
\therefore \dfrac{\sqrt 3}{2}=\dfrac{h}{90}
\therefore h = 45\sqrt 3
m
The angle of elevation of the top of a tower at a distance of
\displaystyle \frac{50\sqrt{3}}{3}
metres from the foot is
\displaystyle 60^{\circ}
. Find the height of the tower
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0%
\displaystyle 50\sqrt{3}
metres
0%
\displaystyle \frac{20}{\sqrt{3}}
metres
0%
-50
metres
0%
50
metres
Explanation
Let the height of the tower be
h
.
In
ΔABC
\tan 60^{0}=\dfrac{AB}{BC}
=\dfrac { h }{ \dfrac { 50\sqrt { 3 } }{ 3 } }
=>\sqrt 3=\dfrac { 3\times h }{ { 50\sqrt { 3 } } }
=\dfrac { 3\times h }{ { 50\sqrt { 3 } \times \sqrt { 3 } } }
=\dfrac { 3\times h }{ { 50\times 3 } }
∴
h=50
m
If from the top of a tower 50 m high, the angles of depression of two objects due north of the tower are respectively
\displaystyle 60^{\circ}
and
\displaystyle 45^{\circ}
, then the approximate distance between the objects is :
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\displaystyle 50\left ( \sqrt{2}-2 \right )m
0%
\displaystyle 50\left ( \sqrt{3}-3 \right )m
0%
31 \ m
0%
None of these
Explanation
Let the two objects be C, D and the height of the tower is
50
m
In
ΔABC
tan60^{0}=\dfrac{AB}{BC}
=\dfrac{50}{x}
\therefore x =\dfrac{ 50}{\sqrt 3}
... {i}
Now, In
ΔABD
tan45^{0}=\dfrac{AB}{BD}
\Rightarrow 1 =\dfrac{50}{x+y}
\Rightarrow x + y = 50
\Rightarrow y=x-50
\Rightarrow y=50 - \dfrac{50}{\sqrt 3}
...[putting the value from {i}]
\Rightarrow y = \dfrac{50}{\sqrt 3}(\sqrt 3−1)
∴ Distance between two objects is
=
\dfrac{50}{\sqrt 3}(\sqrt 3−1)
Rationalising the denominator we get the Distance
=\dfrac{50 \sqrt{3}(\sqrt{3}-1)}{3}
m
Option D is correct.
A man on a cliff observes a boat at an angle of depression of
\displaystyle 30^{\circ}
which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later the angle of depression of the boat is found to be
\displaystyle 60^{\circ}.
Find the total time taken by the boat from the initial point to reach the shore.
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9
min
0%
7
min
0%
10
min
0%
6
min
The angle of elevation of the top of tower as observed from a pint on the horizontal ground is 'x' if we move a distance 'd' towards the foot of the tower the angle of elevation increases to 'y' then the height of the tower is
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\displaystyle \frac{d\tan x\tan y}{\tan y-\tan x}
0%
\displaystyle d(\tan y+\tan x)
0%
\displaystyle d(\tan y-\tan x)
0%
\displaystyle \frac{d\tan x\tan y}{\tan y+\tan x}
Explanation
Let the height of the tower be h.
\displaystyle \therefore
In
\displaystyle \Delta ABC
\displaystyle \tan y=\dfrac{h}{b}
b =
\displaystyle \dfrac{h}{\tan y}
In
\displaystyle \Delta ABD
\displaystyle \tan x=\dfrac{h}{b+d}
\displaystyle \tan x=\dfrac{h}{\dfrac{h}{\tan y}+d}
\displaystyle \dfrac{h}{\tan x}=\dfrac{h}{\tan y}+d
d =
\displaystyle h\left ( \dfrac{1}{\tan x}-\dfrac{1}{\tan y} \right )
d =
\displaystyle h\left ( \dfrac{\tan y-\tan x}{\tan x\tan y} \right )
h =
\displaystyle \dfrac{d\tan x\tan y}{\tan y-\tan x}
The angels of elevation of the top of a vertical tower from two points
30
meter apart and on the same straight line passing through the base of tower are
\displaystyle 30^{\circ}
and
\displaystyle 60^{\circ}
respectively The height of the tower is
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10 m
0%
15 m
0%
\displaystyle 15\sqrt{3}m
0%
30 m
Explanation
In
ΔAPQ
,
\angle PAQ={ 30 }^{ ∘ }
Therefore,
\tan { { 30 }^{ ∘ } } =\dfrac { PQ }{ AQ }
\Rightarrow
\dfrac { 1 }{ \sqrt { 3 } } =\dfrac { h }{ x }
\Rightarrow
x=\dfrac { h }{ \sqrt { 3 } }
............(eqn 1)
Similarly,
ΔPQB
,
\angle PBQ={ 60 }^{ ∘ }
Therefore,
\tan { { 60 }^{ ∘ } } =\dfrac { h }{ 30-x }
\Rightarrow
\sqrt { 3 } =\dfrac { h }{ 30-x }
............(eqn 2)
From equations 1 and 2,
\sqrt { 3 } =\dfrac { h }{ 30-\dfrac { h }{ \sqrt { 3 } } }
\Rightarrow
\sqrt { 3 } \left( 30-\dfrac { h }{ \sqrt { 3 } } \right) =h
\Rightarrow
2h=30\sqrt { 3 }
\Rightarrow
h=15\sqrt { 3 } m
Thus height of the tower is
15\sqrt { 3 } m
.
Hence option C is correct.
The angle of elevation of the top of a tower from a point on the ground
\displaystyle 30^{\circ}
after walking
200
m towards the tower the angle of elevation becomes
\displaystyle 60^{\circ}
. The height of the tower is
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0%
\displaystyle 100\sqrt{3}
m
0%
\displaystyle 200\sqrt{3}
m
0%
100
m
0%
200
m
Explanation
Let
h
be the height of the tower
In
ΔABC
tan 60^{0}=\dfrac{AB}{BC}
=>\sqrt 3=\dfrac{h}{x}
=>x = \dfrac{h}{\sqrt 3}
.......(i)
In
ΔABD
tan 30^{0}=\dfrac{AB}{BD}
=>\dfrac{1}{\sqrt 3}=\dfrac{h}{x+200}
=>x + 200 = \sqrt 3h
=>x = \sqrt 3h - 200
.......(ii)
From (i) and (ii) we get,
\dfrac{h}{\sqrt 3}=\sqrt 3h−200
=>h = 3h - 200 \sqrt 3
=>2h = 200\sqrt 3
h=> 100\sqrt 3
m
=>h = 100 x 1.732
=>h = 173.2
metre
∴ Height of tower =
173.2
m.
The angles of elevation of the top of a vertical tower from two points 30 metres apart and on the same straight line passing through the base of tower are
\displaystyle 30^{\circ}
and
\displaystyle 60^{\circ}
respectively. The height of the tower is
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0%
10 m
0%
15 m
0%
\displaystyle 15\sqrt{3}m
0%
30 m
Explanation
Let the height of the tower is h
In
ΔABC
\tan 60^{0}=\dfrac{h}{x}
=>\sqrt 3=\dfrac{h}{x}
=>x = \dfrac{h}{\sqrt 3}
4 ......(i)
In
ΔABD
\tan 30^{0}=\dfrac{h}{x+30}
=>\dfrac{1}{\sqrt 3} =\dfrac{h}{x+30}
=>x+30 = h\sqrt 3
........(ii)
=>x=h\sqrt 3-30
From (i) and (ii)
\dfrac{h}{\sqrt 3}=h\sqrt 3−30
=>h = 3h - 30\sqrt 3
=>2h=30\sqrt 3
=>h = 15\sqrt 3
So, the height of tower =
15\sqrt 3
m
If the altitude of the sun is at
\displaystyle 60^{\circ}
then the height of the vertical tower that will cast a shadow of length 20 m is
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\displaystyle 20\sqrt{3}m
0%
\displaystyle \frac{20}{\sqrt{3}}m
0%
\displaystyle \frac{15}{\sqrt{3}}m
0%
\displaystyle 40\sqrt{3}m
Explanation
Let the height of the tower be
h
\tan 60^{0}=\dfrac{h}{20}
\Rightarrow \sqrt 3=\dfrac{h}{20}
\Rightarrow h = 20\sqrt 3
m
\Rightarrow h=20\times 1.732
m
\therefore h= 34.64
m
The length of a ladder is exactly equal to the height of the wall it is leaning against. If the lower end of the ladder is kept on a bench of height 3 m. and the bench is kept 9 m. away from the wall, the upper end of the ladder coincides with the top of the wall. The height of the wall is
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11 m
0%
12 m
0%
15 m
0%
18 m
Explanation
AB^2+BC^2=AC^2
(h-2)^2+9^2=h^2
h=15 m
The shadow of a tower is 30 meters when the sun's altitude is
\displaystyle 30^{\circ}
. When the sun's altitude is
\displaystyle 60^{\circ}
then the length of shadow will be
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0%
60 m
0%
15 m
0%
10 m
0%
5 m
Explanation
In fig. AB is the tower , BC is the length of the shadow of tower when altitude is
30^{\circ}
i.e the angle of elevation of the top of the tower from the tip of the shadow is
30^{\circ}
and DB is the length of the shadow, when the angle of elevation is
60^{\circ}
Given, BC = 30 m
In
\triangle ABC
\tan 30^{\circ}= \dfrac {AB}{BC}
\Rightarrow \dfrac 1{\sqrt 3}= \dfrac {AB}{30}
\Rightarrow AB= \dfrac {30}{\sqrt 3} = 10\sqrt 3 m
Now,
In
\triangle ABD
\tan 60^{\circ}= \dfrac {AB}{BD}
\Rightarrow \sqrt 3= \dfrac {10\sqrt 3}{BD}
\Rightarrow BD = 10 m
Option C is correct.
The angle of depression of a car moving with uniform speed towards the building as observed from the top of the building is found to be
30^{\circ}
. The same angle of depression changes to
60^{\circ}
after
12
seconds. How much more time would the car take to reach the base?
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6
sec
0%
8
sec
0%
4
sec
0%
12
sec
Explanation
Let
'h'
be the height of the building.
BC
=x
and CD
=y
In
\triangle ABC,
\Rightarrow \tan { 60° } = \dfrac { h }{ x }
\Rightarrow \sqrt { 3 } = \dfrac { h }{ x }
\Rightarrow h=\sqrt { 3 }x \longrightarrow \left( 1 \right)
In
\triangle ADB,
\Rightarrow \tan { 30° } =\dfrac { h }{ x+y }
\Rightarrow \dfrac { 1 }{ \sqrt { 3 } } =\dfrac { h }{ x+y }
\Rightarrow x+y=\sqrt { 3 } h
From
(1),
\Rightarrow x+y=\sqrt { 3 } \times \sqrt { 3 } x
\Rightarrow x+y=3x
\Rightarrow y=2x
\Rightarrow x=\dfrac{y}{2}
Since
y
is the distance traveled in
12
sec from point D to point C.
so, the distance
x
is travelled in
\dfrac{12}{2}=6
sec.
[\because
distance
x
is half the distance
y]
Hence, the answer is
6
seconds
The angle of elevation of the top of a building from the foot of the tower is
30
and the angle of the elevation of the top of the tower from the foot of the building is
60
. If the tower is
50
m high, then the height of the building is :
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\displaystyle \frac { 50 }{ 3 }
m
0%
\displaystyle \frac { 35 }{ 3 }
m
0%
\displaystyle \frac { 47 }{ 3 }
m
0%
\displaystyle \frac { 52 }{ 3 }
m
Explanation
Let
AB
be the tower and
CD
be the building.
Since the angle of elevation of the top of the
building from the foot of the tower is
30^{0}
.
∴∠CBD=306{0}
Again, the angle of elevation of the top of the tower from the foot of the building is
60^{0}
.
∴∠ADB=60^{0}
In right-angled
ΔABD
, we have
\tan60^{0}=\dfrac{AB}{BD}
⇒\sqrt 3=\dfrac{50}{BD}
⇒BD=\dfrac{50}{\sqrt 3}
........(i)
In right-angled
ΔBDC
, we have
\tan 30^{0}=\dfrac{CD}{BD}
\Rightarrow CD=\dfrac{1}{\sqrt 3}\times \dfrac{50}{\sqrt 3}
....{from {i}}
\Rightarrow CD=\dfrac{50}{3}
m
The angles of elevation of the top of a tower from two points a and b from the base and in the same straight line with it are complementary. The height of the tower is :
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\displaystyle ab
0%
\displaystyle \sqrt { ab }
0%
\displaystyle { a }^{ 2 }{ b }^{ 2 }
0%
None of these
Explanation
Let AB be the tower and C and D be the two points such that BD = a and BC = b.
If
\displaystyle \angle BDA=\theta ,
then
\displaystyle \angle BCA={ 90 }^{ o }-\theta
Now in right-angled
\displaystyle \Delta ABD
,
\displaystyle \frac { AB }{ BD } =\tan { \theta } \Rightarrow \frac { h }{ a } =\tan { \theta }
\displaystyle \therefore \quad h=a\tan { \theta } ......(i)
In right-angled
\displaystyle \Delta ABC
,
\displaystyle \frac { AB }{ BC } =\tan { \left( { 90 }^{ o }-\theta \right) } =\cot { \theta }
\displaystyle \Rightarrow \quad \frac { h }{ b } =\cot { \theta } \Rightarrow h=b\cot { \theta } ......(ii)
multiplying (i) and (ii), we get
\displaystyle { h }^{ 2 }=ab\tan { \theta } \cot { \theta } =ab\tan { \theta } \times \frac { 1 }{ \tan { \theta } } =ab
\displaystyle \therefore \quad h=\sqrt { ab }
Two poles of equal heights are standing opposite to each other on either side of a road, which is
100
metres wide. From a point between them on the road, the angles of elevation of their tops are
30
and
60
. The height of each pole is :
Report Question
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44
m
0%
43.25
m
0%
50
m
0%
40.5
m
Explanation
Let
AB
and
CD
be the two poles and
M
be the point where the angle of elevation is made of
30^{0}
and
60^{0}
respectively.
Let the road
CM
=x
m and
AM
=100-x
m
In right angled
\triangle MCD
, we have
\tan60^{0}=\dfrac{DC}{CM}
\Rightarrow \sqrt 3=\dfrac{h}{x}
\Rightarrow h=x\sqrt 3
(i)
In right angled
\triangle MAB
, we have
\Rightarrow \tan30^{0}=\dfrac{BA}{AM}
\Rightarrow \dfrac{1}{\sqrt 3}=\dfrac{h}{100-x}
\Rightarrow h \sqrt 3=100-x
\Rightarrow (x\sqrt 3)\sqrt 3=100-x
....(from {i})
\Rightarrow 3x=100-x
\Rightarrow 4x=100
\Rightarrow x=25
m
Therefore, height of pole
CD
,
\Rightarrow CD=25\times 1.732
\Rightarrow CD=43.25
m
Thus, height of each pole is
43.25
m.
The angle of elevation of the top of a tower from a point on the ground, which is
30
m away from the foot of the tower is
30
. The height of the tower is :
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0%
\displaystyle 8\sqrt { 3 }
m
0%
\displaystyle 9\sqrt { 3 }
m
0%
\displaystyle 10\sqrt { 3 }
m
0%
\displaystyle 12\sqrt { 3 }
m
The shadow of a vertical tower on level ground increases by
10
metres when the altitude of the sun changes from the angle of elevation
45^0
to
30^0
. Find the height of the tower correct to
one place of decimal.
(take
\displaystyle \sqrt { 3 } =1.732
)
Report Question
0%
13.67\ m
0%
15\ m
0%
18.67\ m
0%
20\ m
Explanation
Let height of tower
CD
be
h
metres
Now in right-angled
\displaystyle \Delta BCD
, we have
\displaystyle \frac { h }{ x } =\tan { { 45 }^{ o } }
\displaystyle \Rightarrow \frac { h }{ x } =1
\Rightarrow h=x
......(i)
Again in right-angled
\displaystyle \Delta ACD
, we have
\displaystyle \frac { h }{ x+10 } =\tan { { 30 }^{ o } }
\displaystyle \Rightarrow \frac { h }{ x+10 } =\frac { 1 }{ \sqrt { 3 } }
\displaystyle \Rightarrow \sqrt { 3 } h=x+10
\displaystyle \Rightarrow \sqrt { 3 } h=h+10
......using (i)
\Rightarrow \displaystyle h\left( \sqrt { 3 } -1 \right) =10
\displaystyle \Rightarrow h=\frac { 10 }{ \sqrt { 3 } -1 } \times \frac { \sqrt { 3 } +1 }{ \sqrt { 3 } +1 }
\displaystyle =\frac { 10\left( \sqrt { 3 } +1 \right) }{ { \left( \sqrt { 3 } \right) }^{ 2 }-{ \left( 1 \right) }^{ 2 } } =\frac { 10\left( 1.732+1 \right) }{ 3-1 }
\displaystyle =5\times 2.732=13.67
m
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle
30
with it. The distance between the foot of the tree to the point where the top touches the ground is
8
m. The height of the tree is:
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\displaystyle 5\sqrt { 3 }
m
0%
\displaystyle 8\sqrt { 3 }
m
0%
\displaystyle 10\sqrt { 3 }
m
0%
\displaystyle 6\sqrt { 3 }
m
Explanation
Let
AB
be the tree.
Let it be broken by the wind at the point
C
and
CB
taking the position
CD
strikes the ground at
D
.
Now,
\displaystyle \angle ADC={ 30 }^{ o }
Let
AC = x
and
CD = y
In right
\displaystyle \Delta ADC
, we have
\displaystyle \frac { AC }{ AD } =\tan { { 30 }^{ o } }
\displaystyle \Rightarrow \frac { x }{ 8 } =\frac { 1 }{ \sqrt { 3 } } \Rightarrow x=\frac { 8 }{ \sqrt { 3 } }
Again
\displaystyle \frac { AD }{ CD } =\cos { { 30 }^{ o } } \Rightarrow \frac { 8 }{ y } =\frac { \sqrt { 3 } }{ 2 } \Rightarrow y=\frac { 16 }{ \sqrt { 3 } }
\displaystyle \therefore
Height of the tree
= (x + y)
metres.
\displaystyle =\frac { 8 }{ \sqrt { 3 } } +\frac { 16 }{ \sqrt { 3 } } =\frac { 24 }{ \sqrt { 3 } } =8\sqrt { 3 }
metres.
From the top of a building h metres high, the angle of elevation of a monument is
45^o
and angle of depression of its foot is
30^o
. The height of the monument is
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0%
\sqrt 3h
0%
\dfrac {(\sqrt 3+1)}{2}h
0%
\dfrac {\sqrt 3-1}{2}h
0%
(\sqrt 3+1)h
Explanation
In
\Delta ADB
AB=\dfrac {h}{tan 30^\circ}
In
\Delta DCF, FC=CD tan 45^\circ
But
AB=CD
\rightarrow\dfrac {h}{tan 30^\circ}=\sqrt 3h
Thus,
FC=\sqrt3h
Total height of monument
=BC+FC
=h+\sqrt 3h
=(\sqrt 3+1)h
From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a
20
m high building are 45 and 60 respectively. The height of the tower is:
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0%
\displaystyle 25\left( \sqrt { 3 } -1 \right)\ m
0%
\displaystyle 20\left( \sqrt { 3 } -1 \right)\ m
0%
\displaystyle 20\ m
0%
\displaystyle 10\ m
A kite is flying at a height of
60
m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is
60
. Find the length of the string, assuming that there is no slack in the string.
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0%
\displaystyle 40\sqrt { 3 }
m
0%
\displaystyle 30\sqrt { 3 }
m
0%
\displaystyle 20\sqrt { 3 }
m
0%
\displaystyle 10\sqrt { 3 }
m
Explanation
Given, the height of the kite
AB
is
60
m and it makes an angle of elevation at point
C
of
60^{0}
.
In right angled
\triangle ABC
, we have
cosec 60^{0}=\cfrac{AC}{AB}
\Rightarrow \cfrac{2}{\sqrt 3}=\cfrac{AC}{60}
\Rightarrow AC=60\times \cfrac{2}{\sqrt 3}
\Rightarrow AC=60\times \cfrac { 2\times \sqrt { 3 } }{ \sqrt { 3 } \times \sqrt { 3 } }
\Rightarrow AC=40\sqrt 3
m
The angle of elevation of the top of a hill at the foot of a tower is
60
and the angle of elevation of top of the tower from the foot of the hill is
30
. If the tower is
50
m high, then the height of the hill is :
Report Question
0%
148
m
0%
150
m
0%
152
m
0%
160
m
Explanation
Let
AB
be the hill and
CD
be the tower.
Since the angle of elevation of the top of the
tower from the foot of the hill is
30^{0}
.
∴∠CBD=30^{0}
Again, the angle of elevation of the top of the hill from the foot of the tower is
60^{0}
.
∴∠ADB=60^{0}
In right-angled
ΔBDC
\tan30^{0}=\dfrac{CD}{BD}
\Rightarrow \dfrac{1}{\sqrt 3}=\dfrac{50}{BD}
⇒BD=50\sqrt3
m .....(i)
In right-angled
ΔABD
, we have
\tan60^{0}=\dfrac{AB}{BD}
\Rightarrow \sqrt 3=\dfrac{AB}{50\sqrt 3}
....
{from (i)}
\Rightarrow AB={\sqrt 3}\times 50\sqrt 3
........(ii)
\Rightarrow AB=150
m
The shadow of a pole standing on a horizontal plane is
a
meters longer when the sun's elevation is
\theta
than when it is
\phi
. The height of the pole will be: {Use
\sin(A-B) = \sin A \cos B - \sin B \cos A
}
Report Question
0%
a\displaystyle\frac{\cos\theta\cdot\cos\phi}{\cos(\theta-\phi)}
meter
0%
a\displaystyle\frac{\sin\theta\sin\phi}{\sin(\phi-\theta)}
meter
0%
a\displaystyle\frac{\sin\theta\cos\phi}{\sin(\theta-\phi)}
meter
0%
a\displaystyle\frac{\sin\phi\cos\theta}{\cos(\theta-\phi)}
meter
Explanation
Let the height of the pole standing on a horizontal plane be
h
Let the length of the shadow be
x
and
(x+a)
, when the sun's elevation is
\phi
and
\theta
respectively.
In
\triangle APQ
,
\dfrac{x}{h}=\cot \phi
\Rightarrow x=h \cot \phi
....(1)
In
\triangle BPQ
,
\dfrac{x+a}{h}=\cot \theta
\Rightarrow (x+a)=h \cot \theta
....(2)
\Rightarrow (x+a)-x=h \cot \theta-h \cot \phi
{using (1)}
\Rightarrow a=h(cot \theta-cot \phi)
\Rightarrow h=\cfrac { a }{ \left( \dfrac { \cos \theta }{ \sin \theta } -\dfrac { \cos \phi }{ \sin \phi } \right) }
\Rightarrow h=a\cfrac { \sin \theta \sin \phi }{ \left( \sin \phi \cos \theta -\cos \phi \sin \theta \right) }
\Rightarrow h=a\cfrac { \sin \theta \sin \phi }{ \sin\left( \phi - \theta \right) }
From the top of house
32
meter high, if the angle of elevation of the top of a tower is
\displaystyle 45^{\circ}
and the angle of depression of the foot of the tower is
\displaystyle 30^{\circ}
, then the height of the tower is
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0%
\displaystyle \frac{32}{\sqrt{3}}(\sqrt{3+1})
meter
0%
\displaystyle 32(\sqrt{3+1})
meter
0%
\displaystyle 32\sqrt3
meter
0%
\displaystyle \frac{32}{3}(\sqrt{3+1})
meter
Explanation
From
\displaystyle \bigtriangleup OBH
\cot 30^{\circ}=\dfrac{BH}{OB}
\Rightarrow \sqrt 3 =\dfrac{BH}{OB}
\Rightarrow \sqrt 3 =\dfrac{BH}{32}
or
\displaystyle BH =32\sqrt{3}
Height of Tower
\displaystyle =OT=OB+BT=32(\sqrt{3}+1)\ meters
An aeroplane flying horizontally at a height of
1.5
km above the ground is observed at a certain point on earth to subtend an angle of
60
. After
15
seconds, its angle of elevation is observed to be
30
. Calculate the speed of the, aeroplane in km/h.
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0%
\displaystyle 240\sqrt { 3 }
km/h
0%
\displaystyle 230\sqrt { 3 }
km/h
0%
\displaystyle 210
km/h
0%
\displaystyle 220
km/h
Explanation
Let
P
and
Q
be the two position of the plane.
Given angles of elevation of the plane in two position
P
and
Q
as
\angle PAB=60^{0}
and
\angle QAC=30^{0}
PB=QC=
15
km
Now in right angled
\triangle ABP
\tan 60^{0}=\dfrac{BP}{AB}
\Rightarrow \sqrt 3=\frac{1.5}{AB}
\Rightarrow AB=\dfrac{1.5}{\sqrt 3}
\Rightarrow (0.5)\sqrt 3
km
Again in right angled triangle
\triangle ACQ
,
\tan30^{0}=\dfrac{QC}{AC}
\Rightarrow \dfrac{1}{\sqrt 3}=\dfrac{1.5}{AB}
\Rightarrow AC=(1.5)\sqrt 3
km
PQ=BC=AC-AB
=(1.5)\sqrt 3-(0.5)\sqrt 3
=(1.5-0.5)\sqrt 3
=\sqrt 3
km
The plane travels
PQ=
\sqrt 3
km in
15
secs
\therefore
Speed of the aeroplane
=
\dfrac{distance}{time}
\Rightarrow \dfrac { \sqrt { 3 } }{ \frac { 15 }{ 3600 } }
\Rightarrow 240 \sqrt 3
km/h
If the angle of elevation of an object from a point 200 meter above the lake is found to be
\displaystyle 30 ^{\circ}
and the angle of depression of its image in the lake is
\displaystyle 45^{\circ}
, then the height of the object above the lake is
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0%
\displaystyle \frac{200\left ( \sqrt{3-1} \right )}{\left ( \sqrt{3+1} \right )}
meter
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\displaystyle \frac{200\left ( \sqrt{3-1} \right )}{\sqrt{3}}
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\displaystyle \frac{200\left ( \sqrt{3+1} \right )}{\sqrt{3}}
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\displaystyle \frac{200\left ( \sqrt{3+1} \right )}{\left ( \sqrt{3+1} \right )}
meter
Explanation
Let OL=h, therefore LO'=H
From
\displaystyle \bigtriangleup PQO',PQ=QO'=QL+LO'
=200+h [OL=LO'=h]
From
\displaystyle \bigtriangleup PQO,\frac{OQ}{PQ}=\tan 30^{\circ}
or
\displaystyle OQ=PQ \tan 30^{\circ}
or
\displaystyle h-200=(200+h)\frac{1}{\sqrt{3}} \therefore h=\frac{200(\sqrt{3}+1)}{\sqrt{3}-1}
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height
h
. At a point on the plane, the angle of elevation of the bottom of the flagstaff is
\displaystyle \alpha
and that of the top of the flagstaff is
\displaystyle \beta
Then the
height of the tower is :
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\displaystyle \frac { h }{ \tan { \alpha } }
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\displaystyle \frac { h\tan { \alpha } }{ \tan { \beta } -\tan { \alpha } }
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\displaystyle \frac { \tan { \beta } }{ h }
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\displaystyle \frac { \tan { \alpha } }{ \tan { \beta } -\tan { \alpha } }
Explanation
Let
BC
be the tower and
CD
be the flagstaff.
The angle of elevation of the bottom of the flagstaff is
\alpha
and that of the top of flagstaff is
\beta
.
Let
h
be the height of the tower
In
\triangle ABC
, we have
\Rightarrow \tan \alpha=\dfrac{BC}{AB}
....{i}
In
\triangle ABD
\Rightarrow \tan \beta=\dfrac{BD}{AB}
\Rightarrow \dfrac{BC+h}{AB}=\dfrac{BD}{AB}
....{ii}
Now dividing {ii} by {i}, we get
\dfrac{BC+h}{BC}=\dfrac{\tan \beta}{\tan \alpha}
\Rightarrow (BC+h)\tan \alpha=BC \tan \beta
\Rightarrow BC(\tan \beta-\tan \alpha)=h \tan \alpha
\Rightarrow BC=\dfrac{h \tan \alpha}{\tan \beta- \tan \alpha}
The angles of elevations of the top of the tower from two points in the same straight line and at a distance of
9 \text{ m}
and
16\text{ m}
from the base of the tower are complementary. The height of the tower is :
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18\text{ m}
0%
16\text{ m}
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10\text{ m}
0%
12\text{ m}
Explanation
From the given figure,
\displaystyle \tan { \theta } =\frac { h }{ 9 }
-------
(1)
\displaystyle \tan { \left( 90-\theta \right) } =\frac { h }{ 16 }
\implies\cot \theta=\dfrac{h}{16}
.......
(2)
Multiplying
(1)
and
(2)
\implies \displaystyle \tan { \theta } \times \cot { \theta } =\frac { { h }^{ 2 } }{ 9\times 16 }
[\because \tan \theta \times \cot \theta=1]
\implies \displaystyle h^2=9\times 16 \\\implies h=\displaystyle \sqrt{9\times 16}\\\implies h=12\text{ m}
\therefore h=12\text{ m}
The angle of elevation of the top of a tower at a distance of
\displaystyle\frac{50\sqrt{3}}{3}
metres from the foot is
60^\circ
. Find the height of the tower.
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50\sqrt{3}\:m
0%
\displaystyle\frac{20}{\sqrt{3}}\:m
0%
-50 m
0%
50 m
Explanation
Let the height of the tower be h metres
\displaystyle\therefore\dfrac{h}{\displaystyle\left(\frac{50\sqrt{3}}{3}\right)}=\tan{60^\circ}=\sqrt{3}
\therefore h=50\:m
The angles of elevation of the top of a tower from two points at the distances of
4\ m
and
9\ m
from the base of the tower and in the same straight line with it are complementary. The height of the tower is :
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8\ m
0%
5\ m
0%
6\ m
0%
4\ m
Explanation
Let AB be the tower and C and D be the two points such that BD =
4 m and BC = 9 m.
If
\displaystyle \angle BDA=\theta ,then\quad \angle BCA={ 90 }^{ o }-\theta
Now in right-angled
\displaystyle \Delta ABD
\displaystyle \frac { AB }{ BD } =\tan { \theta } \Rightarrow \frac { h }{ 4 } =\tan { \theta }
\displaystyle \therefore \quad h=4\tan { \theta } .....(i)
In right-angled
\displaystyle \Delta ABC
\displaystyle \frac { AB }{ BC } =\tan { \theta } \left( { 90 }^{ o }-\theta \right) =\cot { \theta }
\displaystyle \Rightarrow \quad \frac { h }{ 9 } =\cot { \theta } \Rightarrow h=9\cot { \theta } .....(ii)
multiplying (i) and (ii), we get
\displaystyle { h }^{ 2 }=36\tan { \theta } \cot { \theta } =36\tan { \theta } \times \frac { 1 }{ \tan { \theta } } =36\\
\displaystyle \therefore \quad h=\pm \sqrt { 36 } =6m
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