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Some Applications of Trigonometry

Long Answer Type

31. The angles of depression of the top and the bottom of a 9 m high building from the top of a tower are 30° and 60° respectively. Find the height of the tower and the distance between the building and the tower.

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32. From the top of a building 60 m high the angles of depression of the top and the bottom of tower are observed to be 30° and 60°. Find the height of the tower.

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33. As observed from the top of n lighthouse, 100 m high above sea level, the angle of depression of a ship sailing directly towards it, changes from 30° to 60°. Determine the distance travelled by the ship during the period of observation.

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34. From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30° and 45° respectively. Find the height of the hill.

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35. An aeroplane when flying at a height of 3000 m from the ground passes vertically above another aeroplane at an instant when the angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aeroplane at that instant.

Let C and D be the position of two aeroplanes. The height of the aeroplane which is at point D be 3000 m and it passes another aeroplane vertically which is at point C. Let BC = x m. It is also given that the angles of elevation of two planes from the point A on the ground is 45° and 60° respectively.
In right triangle ABC, we have

$tan space 45 degree space equals BC over AB rightwards double arrow space space space space 1 space equals space straight x over AB rightwards double arrow space space space space AB space equals space straight x space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle ABD, we have
Let C and D be the position of two aeroplanes. The height of the aero ]
$tan space 60 degree space equals space BD over AB rightwards double arrow space space square root of 3 equals 3000 over AB rightwards double arrow space space space AB space equals space fraction numerator 3000 over denominator square root of 3 end fraction space space space space space space space space space... left parenthesis ii right parenthesis$
Comparing (i) and (ii), we get

$straight x space equals space fraction numerator 3000 over denominator square root of 3 end fraction$
Hence, vertical distance between the aeroplane
= CD = BD - BC

$equals space 3000 space minus space fraction numerator 3000 over denominator square root of 3 end fraction equals space fraction numerator 3000 square root of 3 minus 3000 over denominator square root of 3 end fraction equals fraction numerator 3000 left parenthesis square root of 3 minus 1 right parenthesis over denominator square root of 3 end fraction equals space fraction numerator 3000 cross times 0.732 over denominator 1.732 end fraction equals 1267.898 space straight M.$

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36. A 7 m long flagstaff is fixed on the top of a tower on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 45° and 30° respectively. Find the height of the tower.

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37. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 metres away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and the width of the river.

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38. From a point 100 m above a lake, the angle of elevation of a stationary helicopter is 30° and the angle of depression of reflection of the helicopter in the lake is 60°. Find the height of the helicopter.

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39. A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height y. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are a and fi respectively. Prove that the height of the tower is

fraction numerator straight y space tan space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction.

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Short Answer Type

40. If the angles of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reflection in the lake is β, prove that the height of the cloud is

straight h fraction numerator left parenthesis tan space straight beta space plus space tan space straight alpha right parenthesis over denominator tan space straight beta space minus space tan space straight alpha end fraction

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