As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depressionof two ships are 30° and 60°. If one strip is exactly behind the other on the same side of thelight-house then the distance between the two ships is :a)25 √3mb)75 √3mc)50 √3md)None of these.Correct answer is option 'C'. Can you explain this answer?

Question

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depressionof two ships are 30&deg; and 60&deg;. If one strip is exactly behind the other on the same side of thelight-house then the distance between the two ships is :a)25 &radic;3mb)75 &radic;3mc)50 &radic;3md)None of these.Correct answer is option 'C'. Can you explain this answer?

Answer

Height of lighthouse =75m
Angles are 30 and 60
Let x be the distance between the ships and y be the distance between the foot of the lighthouse and closer ship.
So tan 60=

Tan 30 =

x =

Answer

Answer

Let's assume that the two ships are at points A and B, with A being closer to the lighthouse and B being farther away.

From the top of the lighthouse, we can draw a line of sight to each ship. Let's call the angle of depression to ship A as angle α and the angle of depression to ship B as angle β.

We know that the height of the lighthouse is 75 m. Since the line of sight from the top of the lighthouse to each ship is perpendicular to the sea level, we can treat the line of sight as the hypotenuse of a right triangle.

For ship A:
In right triangle ALC, where L is the top of the lighthouse and C is ship A, we have:
Angle ALC = α = 30 degrees (given)
Height of the lighthouse, LC = 75 m (given)
We need to find the distance AL.

Using trigonometry, we can use the tangent function:
tan(α) = LC / AL
tan(30 degrees) = 75 m / AL
√3/3 = 75 m / AL
AL = 75 m / (√3/3)
AL = 75√3 m

For ship B:
In right triangle BLD, where L is the top of the lighthouse and D is ship B, we have:
Angle BLD = β = 30 degrees (given)
Height of the lighthouse, LD = 75 m (given)
We need to find the distance BL.

Using trigonometry, we can use the tangent function:
tan(β) = LD / BL
tan(30 degrees) = 75 m / BL
√3/3 = 75 m / BL
BL = 75 m / (√3/3)
BL = 75√3 m

So, the distance AL to ship A is 75√3 m, and the distance BL to ship B is also 75√3 m.

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