Question

The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet the ground at 60°. Find the angle between the sun rays and the ground at the time of longer shadow.

(a) 45°

(b) 30°

(c) 15°

(d) 90°

Answer 1

Correct option is (b) 30°

Let the length of the shadow be x when it makes an angle of 600 at point C at the ground and 3x at an angle of θ at point D.

Let height of flagstaff be h

In right angled ΔABC,

\(\tan 60° = \frac hx\)

⇒ \(h = x \tan 60° \)   .......(1)

In ΔBDA,

\(\tan 3\theta = \frac h{3x}\)

⇒ \(h = 3x\tan \theta\)    .......(2)

From (1) & (2) we get,

x tan 60° = 3x tan θ

⇒ \(\tan \theta = \frac {\tan 60°}3\)

⇒ \(\tan \theta = \frac {\sqrt 3}3\)

⇒ \(\tan \theta = \frac{\sqrt 3 \times \sqrt 3}{3 \times \sqrt 3}\)

⇒ \(\tan \theta = \frac 1{\sqrt 3}\)

⇒ \(\theta = 30°\)