Question

A man in a boat rowing away from a light-house $100m$ high, takes $2$ minutes to change the angle of elevation of the top of the light-house form ${60}^o$ to ${45}^o$. Find the speed of the boat.

• A. $1.627km/hr$
• B. $1.269km/hr$
• C. $1.371km/hr$
• D. $1.523km/hr$

Answer 1

The correct option is B $1.269km/hr$

Height of light-house = 100 m

Time taken to change the angle of elevation from ${60}^{\circ }$ and ${45}^{\circ }$ = 2 minutes

Let the distance from the light house be $d_a$, $d_b$ when the angle of elevation be ${60}^{\circ }$ and ${45}^{\circ }$ respectively
Thus, $\tan \theta$ $=\frac{height}{distancefromlighthouse}$

Thus, $\tan 60=\frac{100}{d_a}$

$\Rightarrow d_a=\frac{100}{\sqrt{3}}$
and, $\tan 45=\frac{100}{d_b}$

$\Rightarrow d_b=100$

$\therefore$ Distance travelled $=100-\frac{100}{\sqrt{3}}$

$=42.2$ m

$Speed=\frac{42.2}{2\times 60}$ $\frac{m}{s}$ {$\because Speed=\frac{distance}{time}$}

To change $\frac{m}{s}$ to $\frac{km}{hr}$ we need to multiply by $\frac{18}{5}$

$\therefore Speed=(\frac{42.2}{2\times 60}\times \frac{18}{5})$ $\frac{km}{hr}$

Hence the $Speed=1.269\frac{km}{hr}$

Option B is correct.