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Some Applications of Trigonometry

Short Answer Type

171.

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of a hill as 30°. Find the distance of the hill from the ship and the height of the hill.

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172.

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60^o. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of the tower is 45^o. Find the height of the tower PQ and the distance PX. $Use square root of 3 space equals 1.73$

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173.

In Given figure, a tower AB is 20 m high and BC, its shadow on the ground, is m $20 square root of 3$ long. Find the sun's altitude.

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174. The angle of elevation of an aeroplane from a point A on the ground 60°. After a flight of 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a constant height of $1500 space square root of 3 space end root space straight m$ , find the speed of the plane in km/hr.

Let P and Q be the two positions of the plane and A be the point of observation. Let ABC be the horizontal line through A. It is given that angles of elevation of the plane in two positions P and Q from a point A are 60° and 30°, respectively.

∴ ∠PAB = 60^o and ∠QAC = 30^o

Also comma space PB space equals space QC space equals space 1500 square root of 3 space straight m In space increment ABP comma space we space have tan space 60 degree space equals space BP 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 space AB space equals space 1500 space straight m In space increment ACQ comma space we space have space tan space 30 degree space equals space QC over AC rightwards double arrow space fraction numerator 1 over denominator square root of 3 end fraction space equals space fraction numerator 1500 square root of 3 over denominator AB end fraction rightwards double arrow space AC space equals space 1500 space straight x space 3 space equals space 4500 space straight m therefore space BC space equals space AC space minus space AB space equals space 4500 space straight m space minus space 1500 space straight m space equals space 3000 space straight m Thus comma space the space plane space travels space 3000 space meters space in space 15 space second. therefore space Speed space space of space the space plane equals space 3000 over 15 equals space 20 space straight m divided by straight s space equals space fraction numerator 200 space straight x space 60 space straight x space 60 space over denominator 1000 end fraction equals 720 space km divided by straight h Hence comma space the space speed space of space the space plane space is space 720 space km divided by straight h.

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Long Answer Type

175. At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30°. The angle of depression of the reflection of the cloud in the lake, at A is 60°. Find the distance of the cloud from A.

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Short Answer Type

176.

The ratio of the height of a tower and the length of its shadow on the ground is √3:1. What is the angle of elevation of the sun?

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177.

A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/h.

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Long Answer Type

178.

The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of the water.

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Short Answer Type

179.

As observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [ Use √3= 1.732 ]

Multiple Choice Questions

180.

The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30^o. The distance of the car from the base of the tower (in m.) is