As observed from the top of a 100m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Step 1: Define variables and set up the first triangle

Let $AB$ be the height of the lighthouse, which is 100 m. Let $C$ be the position of the first ship. The angle of depression to the first ship is $45^\circ$, so the angle of elevation from the ship to the top of the lighthouse, $\angle ACB$, is also $45^\circ$. We use the tangent function in the right-angled triangle $ABC$ to relate the height of the lighthouse to the distance $BC$.

Step 2: Calculate distance to the first ship

Since $\tan 45^\circ = 1$, we can substitute this value into our equation. This allows us to calculate the distance $BC$, which is the distance from the base of the lighthouse to the first ship.

Step 3: Set up the second triangle

Now, let $D$ be the position of the second ship. The angle of depression to the second ship is $30^\circ$, so the angle of elevation from the ship to the top of the lighthouse, $\angle ADB$, is also $30^\circ$. We use the tangent function in the right-angled triangle $ABD$ to relate the height of the lighthouse to the distance $BD$.

Step 4: Calculate distance to the second ship

Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we substitute this value into the equation. This allows us to calculate the distance $BD$, which is the distance from the base of the lighthouse to the second ship.

Step 5: Find the distance between the two ships

The distance between the two ships, $CD$, is the difference between the distance to the second ship ($BD$) and the distance to the first ship ($BC$). We substitute the calculated values to find the final distance.

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