The angles of elevation and depression of the top and bottom of a light house from the top of a building 60 m high are `30^(@)` and `60^(@)` respectively. Find
The difference between the height of the light house and the building.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a building of height 60 m and a lighthouse whose height we need to find. The angle of elevation to the top of the lighthouse from the top of the building is 30 degrees, and the angle of depression to the bottom of the lighthouse is 60 degrees.

### Step 2: Draw a Diagram
Draw a vertical line for the building (60 m) and another vertical line for the lighthouse (height = h). Mark the top of the building as point A, the bottom as point B, the top of the lighthouse as point C, and the bottom of the lighthouse as point D. The horizontal distance from the building to the lighthouse is x.

### Step 3: Set Up the Right Triangles
1. **Triangle ABC** (for the angle of elevation):
- Angle CAB = 30 degrees
- Height of building (AB) = 60 m
- Height of lighthouse (AC) = h
- Therefore, the height difference (BC) = h - 60 m.

2. **Triangle BCD** (for the angle of depression):
- Angle CBD = 60 degrees
- Height of lighthouse bottom (BD) = 60 m (the height of the building).

### Step 4: Apply Trigonometric Ratios
1. For triangle ABC:
\[ \tan(30^\circ) = \frac{BC}{AB} = \frac{h - 60}{x} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{h - 60}{x} \implies x = \sqrt{3}(h - 60) \tag{1} \]

2. For triangle BCD:
\[ \tan(60^\circ) = \frac{BD}{BC} = \frac{60}{x} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{60}{x} \implies x = \frac{60}{\sqrt{3}} \tag{2} \]

### Step 5: Equate the Two Expressions for x
From equations (1) and (2):
\[ \sqrt{3}(h - 60) = \frac{60}{\sqrt{3}} \]

### Step 6: Solve for h
Multiply both sides by \(\sqrt{3}\):
\[ 3(h - 60) = 60 \]
Expanding gives:
\[ 3h - 180 = 60 \]
Adding 180 to both sides:
\[ 3h = 240 \]
Dividing by 3:
\[ h = 80 \text{ m} \]

### Step 7: Find the Difference in Heights
The difference between the height of the lighthouse and the height of the building is:
\[ h - 60 = 80 - 60 = 20 \text{ m} \]

### Final Answer
The difference between the height of the lighthouse and the building is **20 meters**.

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