Some Applications of Trigonometry Test

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Some Applications of Trigonometry Test - 1

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Question 1
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What are the angles of elevation of an aeroplane at a point C from the two places A and B, respectively?

A
60° and 30°

B
60° and 90°

C
30° and 60°

D
90° and 30°

Solution

From point A, angle of elevation = 180° – (90° + 30°) = 60°
And from point B, angle of elevation = 180° – (90° + 60°) = 30°
Question 2
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At a point on the ground, the angle of elevation of a tower of height 'h' is α. On walking 120 m closer to the tower, the angle of elevation becomes β. Which of the following is the correct diagram for this situation?

A

B

C

D

Solution
Question 3
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An aeroplane, which is flying 2 km above the ground, is observed at an elevation of 60°. After 10 seconds, the elevation is observed to be 30°. Which of the following diagrams depicts the given situation?

A

B

C

D

Solution
Question 4
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A 2 m tall observer is 60 m away from an iron pillar. The angle of elevation of the top of the pillar from his eyes is 45°. Which of the following diagrams correctly depicts the given situation?

A

B

C

D

Solution
Question 5
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A ladder is placed against a wall such that the top of the ladder touches it at a height of 20 feet above the ground level. If the ladder makes an angle of 30° with the ground, find its length.

A
20feet

B
feet

C
40 feet

D
10 feet

Solution

Since sin= ^,

sin 30^o = ()
So, the length of the ladder = 40 feet
Question 6
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If the shadow of a 5 ft-tall man is 5ft long, then the angle of elevation of the sun is

A
30°

B
60°

C
45°

D
90°

Solution

Let the height of the man and the length of his shadow be h and l, respectively.
The angle of elevation of the sun is the angle of elevation of the top of the man's head from the tip of the shadow.

Now, tanθ =
… (1)
Also, tan 30° = … (2)
On comparing (1) and (2), we get θ = 30°
Hence, the angle of elevation of sun = 30°
Question 7
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A pole stands vertically on the ground. Two buckets are placed on either side of the pole such that both the buckets and the foot of the pole lie on the same straight line. The angles of elevation of the top of the pole from the two buckets are 45° and 60°, respectively. If the height of the pole is 10 m, find the distance between the two buckets.

A
10 + 10 m

B

C

D

Solution

Let B₁ and B₂ be two buckets.

We have to find the distance between the two buckets, i.e. B₁andB₂.
In ACB₁,

Also, tan
m
And in ACB₂, tan 45° =
Now, B₁B₂ = B₁C + B₂C
B₁B₂ =
B₁B₂ =
Hence, the distance between the two buckets = m
Question 8
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The diameter of a heap of sand in a conical shape is 30 m. The vertical angle is 60°. Find the slant height of the heap of sand.

A
7.5 m

B
30 m

C
20 m

D
15m

Solution

The diameter of a heap of sand in a conical shape is 30 m and the vertical angle is 60°.
So, the semi-vertical angle is 30°.

Diameter (D) = 30 m
Radius = = 15 m
We have to find the slant height of the heap, i.e. AB.
Let us draw a perpendicular AC on BD such that AC divides A into two equal halves.
So, BAC = 30°, BC = 15 m (C is the midpoint of BD)
Now, sin 30° =

Thus, slant height AB = 30 m
Question 9
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The length of the shadow of a pole is times the height of the pole. Find the angle of elevation of the sun.

A
35°

B
60°

C
45°

D
30°

Solution

Let the height of the pole be h.

Then, the length of its shadow will be = h.

We have to find the angle of elevation of the sun, i.e. .

As the angle of elevation of the sun is the angle of elevation of the top of the pole from the tip of the shadow, therefore

tan=
tan =
tan =
We know, tan 30° =
So, = 30°
Question 10
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Two poles are 25 m and 15 m high and the line joining their tops makes an angle of 45° with the horizontal. What is the distance between the poles?

A
10 m

B
m

C
10 m

D
20 m

Solution

Given: The heights of poles h₁ and h₂ are 15 m and 25 m, respectively. The line joining their tops makes an angle of 45° with the horizontal.

We have to find the distance between the poles, i.e. distance CD.
Since BC = 15 m, AD = 25 m and AE = AD – ED
AE = (25 – 15) m ( ED = BC)
AE = 10 m
In ABC, tan 45° =
BE = 10 m (and BE = CD)
Distance between the poles = CD = 10 m
Question 11
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A flagstaff of height h is mounted on the top of a building. From a point on the ground, the angles of elevation of the foot and the top of the flagstaff are and , respectively. If k is the height of the building, then which of the following relations is true?

A
k tan = (h + k)tan

B
k cot = (h + k)tan

C
k tan = (h + k)cot

D
k cot = (h + k)cot

Solution

Since tan = ,
So, in BCD,
tan = BC = …(1)
tan = BC = …(2)
From (1) and (2), … (3)
Also, cot =
So, from (3), k cot = (h + k) cot is a correct relation.
Question 12
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A tower with height h metres casts a shadow of length y metres when the sun's elevation is 45°. Which of the following relationships between h and y is true?

A
h = y

B
y =h

C
y =h

D
h =

Solution

The elevation of the sun's altitude is the angle of elevation of the top of the tower from the tip of the shadow.

Now, tan =
tan 45° =
Thus, h = y is the true relationship.
Question 13
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The height of a conical cap of a joker is 22 inches and the semi-vertical angle is 45°. Find the slant height of the cap.

A
inches

B
44 inches

C
11 inches

D
22 inches

Solution

As cos= , so in a right-angled ABD,

cos 45° =
AB = 22 inches
Question 14
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The length of the shadow of a pole is times the height of the pole. What is the elevation of the sun's altitude?

A
60°

B
30°

C
45°

D
90°

Solution

Let the height of the pole be h.

So, the length of the shadow, = h
The elevation of the sun's altitude is the angle of elevation of the top of the pole from the tip of the shadow.
So, tan q = …. (1)
Comparing (1) and (2), we get q = 30^o.
So, the elevation of the sun's altitude = 30^o
Question 15
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A pole stands vertically on the ground. From a point on the ground which is 30 m away from the foot of the pole, the angle of elevation of the top of the pole is 30^o. Find the height of the pole.

A
m

B
15 m

C
10 m

D
30 m

Solution

Let AB be the pole and the angle of elevation be 30^o.

As tan =

AB = AB = 10m
So, the height of the pole = 10m