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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.

AQ = x m and BQ = y m and the angle of elevation of the top of the tree P from point A and B are 30° and 45° respectively.
i.e., ∠PAQ = 30° and ∠PBQ = 45°

In right triangle
AQP, we have

$tan space 30 degree space equals PQ over AQ rightwards double arrow space fraction numerator 1 over denominator square root of 3 end fraction equals straight h over straight x rightwards double arrow space straight x space equals space square root of 3 space straight h space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle BQP, we have

$tan space 45 degree space equals space PQ over BQ rightwards double arrow space space 1 space equals space straight h over straight y rightwards double arrow space space space straight y space equals space straight h space space space space space space space space space space space space space space space space.... left parenthesis ii right parenthesis$

Adding (i) and (ii), we get

$straight x plus straight y equals square root of 3 space straight h space plus space straight h rightwards double arrow space space 100 space equals space straight h left parenthesis square root of 3 space plus space 1 right parenthesis rightwards double arrow space space space straight h space equals space fraction numerator 100 over denominator square root of 3 plus 1 end fraction straight x fraction numerator square root of 3 minus 1 over denominator square root of 3 minus 1 end fraction space space space space space space space space space equals space fraction numerator 100 left parenthesis square root of 3 minus 1 right parenthesis over denominator left parenthesis square root of 3 right parenthesis squared minus left parenthesis 1 right parenthesis squared end fraction equals fraction numerator 100 left parenthesis 1.732 minus 1 right parenthesis over denominator 3 minus 1 end fraction space space space space space space space space space equals space fraction numerator 100 space straight x space 0.732 over denominator 2 end fraction equals 50 space straight x space 0.732 space space space space space space space space space space equals 36.6 space straight m$

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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.

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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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