Two boats approach a lighthouse in mid-sea from opposite directions. The angles of elevation of the top of the lighthouse from the two boats are 30° and 45° respectively. If the distance between the two boats is 100 m, the height of the lighthouse is:

1. 36.6 m

2. 68.3 m

3. 73.2 m

4. 136.6 m

Let height of lighthouse (CD) be h meters.

Let A and B be the boats approaching lighthouse.

In triangle ACD,

$\Rightarrow \tan 30^\circ = \dfrac{h}{x} \\[1em] \Rightarrow \dfrac{1}{\sqrt3} = \dfrac{h}{x} \\[1em] \Rightarrow x = h\sqrt3.$

In triangle BCD,

$\Rightarrow \tan 45^\circ = \dfrac{h}{y} \\[1em] \Rightarrow 1 = \dfrac{h}{y} \\[1em] \Rightarrow h = y.$

Given,

Distance between the two boats is 100 m.

x + y = 100

$\Rightarrow h\sqrt3 + h = 100 \\[1em] \Rightarrow h(\sqrt{3} + 1) = 100 \\[1em] \Rightarrow h =\dfrac{100}{\sqrt3 + 1} \\[1em] \Rightarrow h = \dfrac{100(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \\[1em] \Rightarrow h = \dfrac{100(\sqrt{3} - 1)}{3 - 1} \\[1em] \Rightarrow h = \dfrac{100(\sqrt{3} - 1)}{2} \\[1em] \Rightarrow h = 50(\sqrt{3} - 1) \\[1em] \Rightarrow h = 50(1.732 - 1) \\[1em] \Rightarrow h = 50(0.732) \\[1em] \Rightarrow h = 36.6 \text{ m}.$

Hence, option 1 is the correct option.