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Some Applications of Trigonometry

Short Answer Type

21. From a point 20 m away from the foot of a tower, the angle of elevation of the top of the tower is 30°, Find the height of the tower.

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22. In figure, what are the angles of depression from the positions O₁ and O₂ of the object at A?

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23. The string of akite is 100 m long and its makes an angle of 60° with the horizontal. Find the height of the kite. Assume that there is no slackness in the string.

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24. In figure, what are the angles of depressions of the top and bottom of a pole from the top of a tower h m high.

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25. A balloon is connected to a meterological ground station by a cable length 215 m inclined at 60° to the horizontal determine the height of the baloon leave the ground. Assume that there is no slack in the cable.

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Long Answer Type

26. There is a small island in the middle of a 100 m wide river and a tall tree stands on the island. A and B are points directly opposite to each other on two banks and in line with the tree. If the angles of elevation of the top of the tree from P and Q are respectively 30° and 45°, find the length of the tree.

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27. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 15 seconds the angle of elevation changes to 30°. If the jet plane is flying at a constant height of $bold 1500 square root of bold 3 bold space bold m$ find the speed of the jet plane.

Let A be the point of observation, C and E be the two points of the plane. It is given that after 15 seconds angle of elevation changes from 60° to 30°.

i.e., ∠BAC = 60° and ∠DAE = 30°. It is also given that height of the jet plane is 1500

square root of 3 space straight m

straight i. straight e. space space space space space space space space space space space space CB space equals space 1500 square root of 3

[Since jet plane is flying at constant height, therefore, CB = ED =

1500 square root of 3 space straight m right square bracket

In right triangle ABC, we have

tan space 60 degree space space equals space BC over AB rightwards double arrow space square root of 3 space equals space fraction numerator 1500 square root of 3 over denominator AB end fraction rightwards double arrow AB space equals space fraction numerator 1500 square root of 3 over denominator square root of 3 end fraction rightwards double arrow space AB space equals space 1500 space straight m space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis

In right triangle ADE, we have

tan space 30 degree space equals space DE over AD rightwards double arrow space space space fraction numerator 1 over denominator square root of 3 end fraction equals fraction numerator DE over denominator AB plus BD end fraction rightwards double arrow space space space fraction numerator 1 over denominator square root of 3 end fraction equals fraction numerator 1500 square root of 3 space over denominator AB plus BD end fraction rightwards double arrow space AB space plus space BD space equals space 1500 square root of 3 cross times square root of 3 rightwards double arrow space space AB space plus space BD space equals space 4500 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis ii right parenthesis

Putting the value of (i) in (ii), we get
1500 + BD = 4500
⇒ BD = 3000
∵ Distance travelled in 15 sec
= CE = BD = 3000 metres,

Now, speed of plane (m/s) = $3000 over 15 equals 200 space straight m divided by straight s$

and speed of plane (km/h) = $space space 200 over 1000 cross times 3600$
= 720 km/hr

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28. The angles of elevation of the top of a tower from two points at a distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is

square root of ab

metres.

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29. An aeroplane at an altitude of 200 metres observes the angles of depression of opposite points on the two banks of a river to be 45° and 60°. Find the width of the river.

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30. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. At a point Y, 40 m vertically above X, the angle of elevation is 45°. Find the height of the tower PQ and the distance XQ.

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