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Some Applications of Trigonometry

Long Answer Type

61. A statue 1.46 m. tall stands on the top of a pedestal. From a point on the ground the angle of elevation of the top of statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45° find the height of the pedestal.

Let hetght of the pedestal BD be h metres, and angle of elevation of C and D at a point A on the ground be 60° and 45° respectively.

It is also given that the height of tine statue CD be 1.6 m
i.e., ∠CAB = 60°,
∠DAB = 45° and CD = 1.6m
In right triangle ABD, we have

$tan space 45 degree space equals space BD over AB rightwards double arrow space space 1 space equals space straight h over AB rightwards double arrow space space space space AB space equals space straight h space space space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle ABC, we have
Let hetght of the pedestal BD be h metres, and angle of elevation of

$tan space 60 degree space equals space BC over AB rightwards double arrow space space space square root of 3 space equals space fraction numerator BD plus DC over denominator AB end fraction rightwards double arrow space space space square root of 3 equals fraction numerator straight h plus 1.46 over denominator AB end fraction rightwards double arrow space space space Ab space equals space fraction numerator straight h plus 1.46 over denominator square root of 3 end fraction space space space space space space space space space space space space space space space space space.... left parenthesis ii right parenthesis$
Comparing (i) and (ii), we get

$straight h equals space fraction numerator straight h plus 1.46 over denominator square root of 3 end fraction rightwards double arrow space space space square root of 3 space straight h space equals space straight h plus 1.46 rightwards double arrow space space space straight h left parenthesis square root of 3 minus 1 right parenthesis space equals space 1.46 rightwards double arrow space space space space straight h space equals space fraction numerator 1.46 over denominator square root of 3 minus 1 end fraction straight x fraction numerator square root of 3 plus 1 over denominator square root of 3 plus 1 end fraction rightwards double arrow space space space space straight h space equals space fraction numerator 1.46 open parentheses square root of 3 plus 1 close parentheses over denominator 3 minus 1 end fraction rightwards double arrow space space space space straight h space equals space 0.73 space open parentheses square root of 3 plus 1 close parentheses$

Hence, the height of pedestal
= 0.73 (1.73 + 1) = 0.73 × 2.73
= 1.929 m. = 2 m. (approx)

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62. The angle of elevation of a jet fighter 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 is flying at a speed of 720 km/hour, find the constant height at which the jet is flying.

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63. A man on the deck of a ship 14 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

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Short Answer Type

64. The height of a tower is 10 m. Calculate the height of its shadow when Sun's altitude is 45°.

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65. In the following figure, what are the angles of depression from the observing positions O₁ and O₂ of the object at A?

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66. Find the angle of elevation of the Sun's altitude when the height of shadow of a vertical pole is equal to its height.

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67. In figure, what are the angles of depression of depression of the top and bottom of h m tall building from the top of multistoryed building.

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68.

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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69. In figure, what are the angles of depression from the positions O₁ and O₂ of the object at A?

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70.

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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