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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.

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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.

Let D be the position of the man and AB be the water level and AB be the cliff. The angles of elevation of the top and depression of the base is 60° and 30.

straight i. straight e. space space space space space angle BDE space space space space equals space 60 degree space and space angle DEA space equals space 30 degree Let space space space space space space space space space BE space space space space space space equals space straight h space and space AC space space DE space equals straight x Now comma space space In space increment space BDE space tan space 60 degree space equals space BE over DE rightwards double arrow space space space space space space space space space space square root of 3 space space space space equals space space straight h over straight x rightwards double arrow space space space space space space space space space space space straight x space space space space equals space space fraction numerator straight h 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 space... left parenthesis straight i right parenthesis In space increment ADE comma space tan space 30 degree space equals space AE over DE rightwards double arrow space space space space space space space space space fraction numerator 1 over denominator square root of 3 end fraction equals 14 over straight x rightwards double arrow space space space space space space space space space space space straight x space space equals space 14 square root of 3 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 h over denominator square root of 3 end fraction equals 14 square root of 3 rightwards double arrow space space space straight h space equals space 14 square root of 3 plus square root of 3 equals 42 space straight m

Hence, Height of the cliff.

= h + 14 = 42 + 14 = 56 m

And space distance space straight x space equals space fraction numerator straight h 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 space space space space equals space fraction numerator 42 over denominator square root of 3 end fraction equals fraction numerator 14 square root of 3 cross times square root of 3 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 space space space space equals space 14 square root of 3 equals 14 space straight x space 1.732 space space space space space space space space space space space space space space space space space space space space space equals space 24.25 space straight m

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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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