Question

A lady on the top of a vertical lighthouse observes a bus moving at a uniform speed coming directly towards it. If the angle of depression changes from ${30}^{\circ }$ to ${45}^{\circ }$ in 12 minutes, how soon after this will the bus reach the lighthouse? Assume $\surd$3 = 1.732.

• A. 14 min. 35 sec
• B. 15 min. 49 sec
• C. 16 min. 23 sec
• D. 18 min. 5 sec
• E. Cannot be determined

Answer 1

The correct option is C

16 min. 23 sec

Let AB be the tower and C & D be the two positions of the bus.

So, $\angle$ACB = ${45}^{\circ }$, $\angle$ADB = ${30}^{\circ }$ and AB = h, CD = x, AC = y

$\frac{AB}{AC}$ = tan ${45}^{\circ }$ = 1 $\Rightarrow$ y = h

$\frac{AB}{AD}$ = tan ${30}^{\circ }$ = $\frac{1}{\sqrt{3}}$

$\Rightarrow$ x + y = $\sqrt{3}$ h

We get x = ($\sqrt{3}$- 1) h is covered in 12 min

So, h will be covered in$\frac{12}{(\sqrt{3}-1)h}$ x h = 16 min 23 sec.