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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.

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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

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

Let AB be the surface of the lake and P be the position of the observer h metres above the lake. Let C be the cloud and C be the reflection in the cloud, then CB = C'B. It is also given that the angle of elevation of cloud from a point h m above a lake is α and angle of depression of its reflection is β.

i.e., ∠CPQ = α
and ∠QPC' = β.
Let CQ = x m
Then
CB = CQ + BQ
= CQ + PA
= x + h
Let AB be the surface of the lake and P be the position of the observ
In right triangle PQC, we have

$tan space straight alpha space equals space CQ over PQ rightwards double arrow space tan space straight alpha space equals space fraction numerator straight x over denominator space PQ end fraction rightwards double arrow space PQ space equals space fraction numerator straight x over denominator tan space straight alpha end fraction space space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle PQC', we have

$tan space straight beta space space space space space space space space space space space equals space space space fraction numerator QC apostrophe over denominator PQ end fraction rightwards double arrow space space space space space space tan space straight beta space space equals space fraction numerator BQ plus BC apostrophe over denominator PQ end fraction rightwards double arrow space space space space space space tan space straight beta space space equals space fraction numerator straight h plus straight x plus straight h over denominator PQ end fraction rightwards double arrow space space space space space space tan space straight beta space space equals space fraction numerator straight x plus 2 straight h over denominator PQ end fraction rightwards double arrow space space space space space space PQ space space equals space fraction numerator straight x plus 2 straight h over denominator tan space straight beta end fraction space space space space space space space space space space space space space space space space space space space space space... left parenthesis ii right parenthesis$
Comparing (i) and (ii), we get

$fraction numerator straight x over denominator tan space straight alpha end fraction space equals space fraction numerator straight x plus 2 straight h over denominator tan space straight beta end fraction rightwards double arrow space space space space space space space space space space straight x space tan space straight beta space equals space straight x space tan space straight alpha space plus space 2 straight h space tan space straight alpha rightwards double arrow space space space space space space straight x space tan space straight beta space minus space straight x space tan space straight alpha space equals space 2 straight h space tan space straight alpha rightwards double arrow space space space space straight x space equals space fraction numerator 2 straight h space tan space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction$
Hence, the height of the cloud

$space space BC space equals space straight x space plus space straight h space space space space space space space equals space fraction numerator 2 straight h space tan space straight alpha over denominator tan space straight beta space minus tan space straight alpha end fraction plus straight h equals fraction numerator 2 straight h space tan space straight alpha space plus straight h space left parenthesis tan space straight beta space minus space tan space straight alpha right parenthesis over denominator tan space straight beta space minus space tan space straight alpha end fraction equals fraction numerator 2 straight h space tan space straight alpha plus straight h space tan space straight beta space minus space straight h space tan space straight alpha over denominator tan space straight beta minus space tan space straight alpha end fraction equals space fraction numerator straight h space tan space straight beta space plus space straight h space tan space straight alpha over denominator tan space straight beta space minus space tan space straight alpha end fraction equals space fraction numerator straight h space left parenthesis tan space straight beta space plus space straight h space tan space straight alpha right parenthesis over denominator tan space straight beta space minus space tan space straight alpha end fraction$
Hence, the height of the cloud is

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

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