Question

Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are 60° and 45°, respectively. If the distance between the ships is `100 ((1 + sqrt(3))/sqrt(3))` m, then find the height of the lighthouse.

Answer 1

Given: From the top of a lighthouse of height h, the angles of depression to two ships on opposite sides are 60° and 45°. The distance between the ships is `100((1 + sqrt(3))/sqrt(3))` m.

Step-wise calculation:

1. Let the horizontal distances from the foot of the lighthouse to the ships be x (for 60°) and y (for 45°).

Then the distance between ships = x + y.

2. Using tan(angle of elevation) = `"Opposite"/"Adjacent"`:

For 60°: `tan 60^circ = h/x` 

⇒ `sqrt(3) = h/x` 

⇒ `x = h/sqrt(3)`

For 45°: `tan 45^circ = h/y` 

⇒ `1 = h/y` 

⇒ y = h

3. So `x + y = h/sqrt(3) + h` 

= `h(1 + 1/sqrt(3))` 

= `h((sqrt(3) + 1)/sqrt(3))`

4. Set this equal to the given distance:

`h((sqrt(3) + 1)/sqrt(3)) = 100((sqrt(3) + 1)/sqrt(3))`

5. Cancel the common factor `(sqrt(3) + 1)/sqrt(3)` (nonzero) to get h = 100.

The height of the lighthouse is 100 m.