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Height and Distance
Height and Distance
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Question

A spherical gas balloon of radius 16 meter subtends an angle 60∘^\circ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75∘^\circ. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

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Solution

O →\to centre of sphere P, Q →\to point of contact of tangents from A Let T be top most point of balloon & R be foot of perpendicular from O to ground. From triangle OAP, OA = 16cosec30∘^\circ = 32 From triangle ABO, OR = OA sin75∘^\circ = 32(3+1)2232{{\left( {\sqrt 3 + 1} \right)} \over {2\sqrt 2 }} So level of top most point = OR + OT =8(6+2+2) = 8\left( {\sqrt 6 + \sqrt 2 + 2} \right)

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