The angles of depression of two ships from the top of a lighthouse are `45^@ and 30^@`. If the ships are 120 m apart, then find the height of the lighthouse.

Answer

To find the height of the lighthouse given the angles of depression to two ships and the distance between the ships, we can follow these steps:

### Step 1: Understand the Problem
We have a lighthouse and two ships. The angles of depression from the top of the lighthouse to the two ships are given as \(45^\circ\) and \(30^\circ\). The distance between the two ships is 120 meters. We need to find the height of the lighthouse.

### Step 2: Set Up the Diagram
Let:
- \(H\) = height of the lighthouse
- \(X\) = horizontal distance from the base of the lighthouse to the first ship (the one at \(45^\circ\))
- \(X + 120\) = horizontal distance from the base of the lighthouse to the second ship (the one at \(30^\circ\))

### Step 3: Apply Trigonometric Ratios
Using the tangent function, we can set up equations based on the angles of depression.

1. For the first ship (angle of depression \(45^\circ\)):
\[ \tan(45^\circ) = \frac{H}{X} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ H = X \quad \text{(Equation 1)} \]

2. For the second ship (angle of depression \(30^\circ\)):
\[ \tan(30^\circ) = \frac{H}{X + 120} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{H}{X + 120} \]
Rearranging gives:
\[ H = \frac{(X + 120)}{\sqrt{3}} \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we know \(X = H\). Substitute \(X\) in Equation 2:
\[ H = \frac{(H + 120)}{\sqrt{3}} \]

### Step 5: Solve for H
Multiply both sides by \(\sqrt{3}\):
\[ H \sqrt{3} = H + 120 \]
Rearranging gives:
\[ H \sqrt{3} - H = 120 \]
Factoring out \(H\):
\[ H(\sqrt{3} - 1) = 120 \]
Thus:
\[ H = \frac{120}{\sqrt{3} - 1} \]

### Step 6: Rationalize the Denominator
To rationalize the denominator:
\[ H = \frac{120(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{120(\sqrt{3} + 1)}{3 - 1} = \frac{120(\sqrt{3} + 1)}{2} \]
This simplifies to:
\[ H = 60(\sqrt{3} + 1) \]

### Final Answer
The height of the lighthouse is:
\[ H = 60(\sqrt{3} + 1) \text{ meters} \]

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