A lighthouse is visible just above the horizon at a certain station at the sea level. Distance between the station and the lighthouse is 60 km. The height of the lighthouse isa)242.28 mb)4 mc)287.5 md)5.4 mCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

Given, Distance between the station and the lighthouse = 60 km

Let the height of the lighthouse be h.

We know that the earth is curved and so the line connecting the observer and the lighthouse is a tangent to the earth's surface at the station.

From the figure, we can observe that:

- The line joining the observer and the top of the lighthouse makes a right-angled triangle with the tangent to the earth's surface at the station and the radius of the earth.
- The distance of the station from the point where the tangent touches the earth's surface is equal to the radius of the earth.
- The distance of the observer from the point where the tangent touches the earth's surface is equal to the radius of the earth plus the height of the observer.

Using the above observations, we can write:

tan θ = h / (R + d)

where θ is the angle of elevation of the top of the lighthouse from the observer, R is the radius of the earth, and d is the height of the observer above the surface of the earth.

Also, we have:

tan Φ = h / d

where Φ is the angle of depression of the observer from the top of the lighthouse.

From the figure, we can see that θ + Φ = 90°.

Substituting the value of tan Φ in the above equation, we get:

tan θ = h / (R + (h / tan Φ))

tan θ = h / (R + (h / (tan(90° - θ))))

tan θ = h / (R + (h / cot θ))

tan θ = h / (R / cot θ + h)

tan θ = h / (R cos θ + h sin θ)

tan θ = h / R (1/cos θ + h/R sin θ)

Let d = R cos θ be the distance of the observer from the point where the tangent touches the earth's surface.

Then, we have:

tan θ = h / (R + d)

tan θ = h / (R + R cos θ)

tan θ = h / R (1 + cos θ)

Solving for h, we get:

h = R tan θ / (1 + cos θ)

Substituting the given values, we get:

h = 6371 km tan 0.016667° / (1 + cos 0.016667°)

h = 242.28 m

Therefore, the height of the lighthouse is 242.28 m.

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