Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the ships are 45° and 60° respectively. If the lighthouse is 81 m high, then the distance between two ships is:

Correct option is C

​Given:
- Height of the lighthouse (BD) = 81 m
- Angle of elevation from the first ship (∠BAD) = 45°
- Angle of elevation from the second ship (∠BCD) = 60°

Solution:

Let the lighthouse be of height BD = 81 m, and let A and C be the positions of the two ships.

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Then, ∠BAD = 45° and ∠BCD = 60°.

In △ABD, we have:

$$tan 45° =\frac{ BD }{BA}\\\ \\ 1 = \frac{81 }{BA}\\\ \\ BA = 81 m$$​​
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In △BCD, we have:
​$$tan 60° =\frac{ BD}{BC}\\\ \\√3 = \frac{81}{ BC}\\\ \\ BC = \frac{81 }{√3}\\\ \\ BC = 81 \times \frac{√3 }{ 3} = 27√3$$​
Now, the distance between the two ships, AC = BA + BC
= 81 + 27√3

 = $$\frac{81(1 + \sqrt3)}{\sqrt3}$$

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Then the distance between two ships = = $$\frac{81(1 + \sqrt3)}{\sqrt3}$$