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

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25 (3 − 1) m

50 (3 − 1) m

75 (3 − 1) m

100 (3 − 1) m

answer is B.

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Detailed Solution

Given that two boats approach a light house in mid-sea from opposite directions.
We have to find the height of the light house.
The value for the

tan 45 ° = 1

Suppose,

B D = h

be the height of the lighthouse and

B C = x

Then, by using the figure geometry,
Question Image

In triangle BDC,

tan 45 ° = B D B C ⇒ 1 = h x ⇒ x = h In triangle D A B,

tan 30 ° = D B B A ⇒ 13 = h 100 − x ⇒ 3 h = 100 − h ⇒ (3 + 1) h = 100

So, on further solving the above expression,

⇒ h = 100 (3 + 1) ⇒ h = 100 (3 + 1) × (3 − 1) (3 − 1) ⇒ h = 50 (3 − 1) m

So, the height of the lighthouse is

50 (3 − 1) m

.
The required height is

50 (3 − 1) m

.
Therefore, the correct option is 2.

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