Question

Two boats are sailing in the sea on either side of a lighthouse. At a particular time the angles of depression of the two boats, as observed from the top of the lighthouse are 45${}^{\circ }$ and 30${}^{\circ }$ respectively. If the lighthouse is 100 m high, find the distance between the two boats.

• A. 273.2 m
• B. 172.4 m
• C. 380 m
• D. 220.2 m

Answer 1

Answer: (A)

273.2 m

Let A and B be the position of two boats (figure)
Let PM be the lighthouse such that PM $=100m$
Let AM $=x$ and BM $=y$
In $\Delta APM,\frac{AM}{PM}=\cot {30}^{\circ }$
$\Rightarrow \frac{x}{100}=\sqrt{3}\Rightarrow x=100\sqrt{3}m$ $[\because \cot {30}^o=\sqrt{3}]$
Similarly, in $\Delta$ BPM,
$\frac{BM}{PM}=\cot {45}^{\circ }\Rightarrow \frac{y}{100}=1\Rightarrow y=100m$ $[\because \cot {45}^o=1]$

$\therefore$ Required distance, $AB=x+y$

$\Rightarrow 100\sqrt{3}+100=100(\sqrt{3}+1)m=273.2m$

Hence, the distance between the two boats is $273.2m$