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Some Applications of Trigonometry

Short Answer Type

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11)

Fig. 9.11

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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. Find the height of the tree.

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A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

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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°. Find the height of the tower.

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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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Long Answer Type

6. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° from 60°, as he walks towards the building. Find the distance he walked towards the building.

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Short Answer Type

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

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Long Answer Type

8.
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Let height of the pedestal BD be h metres, and angle of elevation of C and D at a point A on the ground be 60° and 45° respectively.
It is also given that the height of the statue CD be 1.6 m
i.e., ∠CAB = 60°,
∠DAB = 45° and CD = 1.6m
In right triangle ABD, we have

$tan space 45 degree space equals space BD over AB rightwards double arrow space space space 1 space equals space straight h over AB rightwards double arrow space space space AB thin space equals space straight h space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle ABC, we have

$tan space 60 to the power of degree space equals space BC over AB rightwards double arrow space space square root of 3 space equals space fraction numerator BD plus DC over denominator AB end fraction rightwards double arrow space space square root of 3 space equals space fraction numerator straight h plus 1.6 over denominator AB end fraction rightwards double arrow space space AB space equals space fraction numerator straight h plus 1.6 over denominator square root of 3 end fraction space space space space space space space space... left parenthesis ii right parenthesis$
Let height of the pedestal BD be h metres, and angle of elevation of
Comparing (i) and (ii), we get
$Let height of the pedestal BD be h metres, and angle of elevation of$ $straight h space equals space fraction numerator straight h space plus 1.6 over denominator square root of 3 end fraction rightwards double arrow space space space square root of 3 space end root space straight h space equals space straight h space plus space 1.6 rightwards double arrow space space space square root of 3 space straight h space minus space straight h space equals space 1.6 rightwards double arrow space space space straight h left parenthesis square root of 3 minus 1 right parenthesis space equals space 1.6 rightwards double arrow space space space straight h space equals space fraction numerator 1.6 over denominator square root of 3 minus 1 end fraction straight x fraction numerator square root of 3 plus 1 over denominator square root of 3 plus 1 end fraction rightwards double arrow space space space straight h space equals space fraction numerator 1.6 left parenthesis square root of 3 plus 1 right parenthesis over denominator 3 minus 1 end fraction rightwards double arrow space space space straight h space equals space 0.8 open parentheses square root of 3 plus 1 close parentheses$
Hence, the height of pedestal $equals space 0.8 space left parenthesis square root of 3 plus 1 right parenthesis straight m$

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The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

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Short Answer Type

10.

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

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