From the top of a lighthouse 70 m high with its base at sea level, the angle of depression of a boat is $15^{\circ}$. The distance of the boat from the foot of the lighthouse is :

• A

  $70(2-\sqrt{3}) \mathrm{m}$
• Correct answer
  B

  $70(2+\sqrt{3}) \mathrm{m}$
• C

  $70(3-\sqrt{3}) \mathrm{m}$
• D

  $70(3+\sqrt{3}) \mathrm{m}$

Solution

AB is a lighthouse. Let $\mathrm{AC}=x \mathrm{~m}$. $\begin{aligned} & \tan 15^{\circ}=\frac{70}{x} \\ & \frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{70}{x}\end{aligned}$ $x=\frac{70(\sqrt{3}+1)}{\sqrt{3}-1}$ $=\frac{70(\sqrt{3}+1)}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{70 \times 2(2+\sqrt{3})}{2}$ $\Rightarrow \quad x=70(2+\sqrt{3}) \mathrm{m}$.

Hence, the correct answer is option: 'B'.