Question

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

(1) 8( 2 + 2√3 + √2)

(2) 8(√6 + √2 + 2)

(3) 8(√2 + 2 + √3)

(4) 8(√6 - √2 + 2)

Answer 1

Correct option (2) 8(√6 + √2 + 2)

O → centre of sphere 

P,Q → 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° = 32 

From triangle ABO, OR = OA sin75° = 32 \(\frac{(\sqrt{3} + 1)}{2\sqrt{2}}\) 

So level of top most point = OR + OT 

= 8\((\sqrt{6} + \sqrt{2} +2)\)