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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.

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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.

Let the position of the aeroplane be A, B and C be two points on the two banks of a river such that the angles of depression at B and C are 45° and 60° respectively. Let BD = x m, y m andAD = 200 m.

In right triangle ABD, we have

tan space 45 degree space space equals space AD over BD rightwards double arrow space space space space 1 space space equals space 200 over straight x rightwards double arrow space space space space space straight x space equals space 200 space space straight m space space space space space space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis

In right triangle ACD, we have

tan space 60 degree space space equals space AD over CD rightwards double arrow space space space square root of 3 space equals space 200 over straight y rightwards double arrow space space space space straight y space equals space fraction numerator 200 over denominator square root of 3 end fraction space space space space space space space space space space space space space space space space space... left parenthesis ii right parenthesis

Adding (i) and (ii), we get

space space space space straight x plus straight y space equals space 200 space plus space fraction numerator 200 over denominator square root of 3 end fraction rightwards double arrow space space space space BC space equals space fraction numerator 200 square root of 3 plus 200 over denominator square root of 3 end fraction space straight m space space space space space space space space space space space space space space equals space fraction numerator 200 open parentheses square root of 3 plus 1 close parentheses over denominator square root of 3 end fraction space straight m space space space space space space space space space space space space space space space space equals space 315.4 space straight m

Hence, the width of the river is 315.4 metres.

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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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