Question

A man on the top of the light house observes a boat comming towards it. lf it takes $10$ minutes forthe angle of depresion to change from ${30}^0$ to ${45}^0$, the time in minutes taken by the boat to reach the shore is

• A. $13.66$
• B. $23.66$
• C. $33.66$
• D. $44.66$

Answer 1

Answer: (B)

$23.66$

Let the height of light house be $h$

Let the point at which angle of depression is ${30}^0$ is $A$ and the point at which angle of depression is ${45}^0$ is $B$

Let the foot of light house be $o$

Let $OB=x$ and $AB=y$

We have $tan\left(30\right)=\frac{1}{\sqrt{3}}=\frac{h}{x+y}$ and $tan\left(45\right)=1=\frac{h}{x}$

So we get $y=(\sqrt{3}-1)x$

Let the speed of boat be $v$

Given $y=v\times 10$ , Let the time taken by boat to travel distance $OA$ be $t$

Therefore,

$OA=x+y=vt$

$\Rightarrow v\times 10(1+\frac{1}{\sqrt{3}-1})=vt$

$\Rightarrow t=5(3+\sqrt{3})=23.66$