JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Heights and Distances

A spherical gas balloon of radius 16 meter subtends an angle

6 0^{\circ}

at the eye of the observer A while the angle of elevation of its center from the eye of A is

7 5^{\circ}

. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

Answer: (B)

Solution

The balloon subtends

6 0^{\circ}

at the eye, so the line of sight to the centre

O

makes

3 0^{\circ}

with each tangent line of sight. The radius drawn to a point of tangency is perpendicular to that tangent, so for the eye at

A

sin 3 0^{\circ} = \frac{16}{A O} \Rightarrow A O = 32 m

The centre stands above the eye level by

A O sin 7 5^{\circ} = 32 \cdot \frac{6 + 2}{4} = 8 (6 + 2) m

The topmost point of the balloon lies one radius above the centre, so its height above the observer's eye is

8 (6 + 2) + 16 = 8 (6 + 2 + 2)

JEE Main · 2021 · 25 Jul 2021 · Shift 1 · Single correct option