The angles of elevation oand depression of the top and bottom of a lighthouse from the top of a building, 60 m high, are `30^(@)` and `60^(@)` respectively. Find
(i) the difference between the heights of the lighthouse and the building.
(ii) distance between the lighthouse and the building.

Answer

To solve the problem step by step, we will use trigonometric concepts related to angles of elevation and depression.

### Given:
- Height of the building (AB) = 60 m
- Angle of elevation to the top of the lighthouse (CD) = 30°
- Angle of depression to the bottom of the lighthouse (DE) = 60°

### Step 1: Identify the points
Let's define the points:
- A: Top of the building
- B: Bottom of the building (ground level)
- C: Top of the lighthouse
- D: Bottom of the lighthouse (ground level)
- E: Point directly below the top of the lighthouse on the ground

### Step 2: Set up the right triangles
From point A (top of the building), we have:
- Triangle ACD for the angle of elevation (30°)
- Triangle ABE for the angle of depression (60°)

### Step 3: Calculate the horizontal distance (BD)
Using the angle of elevation (30°):
- In triangle ACD:
\[ \tan(30°) = \frac{CD}{BD} \]
Let BD = x and CD = y (height of the lighthouse).
\[ \tan(30°) = \frac{1}{\sqrt{3}} \Rightarrow \frac{y}{x} = \frac{1}{\sqrt{3}} \Rightarrow y = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

### Step 4: Calculate the height of the lighthouse (CD)
Using the angle of depression (60°):
- In triangle ABE:
\[ \tan(60°) = \frac{AB}{BE} \]
Here, AB = 60 m (height of the building) and BE = x (horizontal distance).
\[ \tan(60°) = \sqrt{3} \Rightarrow \frac{60}{x} = \sqrt{3} \Rightarrow x = \frac{60}{\sqrt{3}} \quad \text{(Equation 2)} \]

### Step 5: Substitute x in Equation 1
Now, substitute the value of x from Equation 2 into Equation 1:
\[ y = \frac{\frac{60}{\sqrt{3}}}{\sqrt{3}} = \frac{60}{3} = 20 \text{ m} \]
Thus, the height of the lighthouse (CD) is 20 m.

### Step 6: Calculate the difference in height
The difference between the height of the lighthouse and the building:
\[ \text{Difference} = CD - AB = 20 \text{ m} - 60 \text{ m} = -40 \text{ m} \quad \text{(This indicates the lighthouse is shorter)} \]

### Step 7: Calculate the distance between the lighthouse and the building
Using the value of x from Equation 2:
\[ BD = \frac{60}{\sqrt{3}} \approx 34.64 \text{ m} \]

### Final Answers:
(i) The difference between the heights of the lighthouse and the building is **40 m** (the lighthouse is shorter).
(ii) The distance between the lighthouse and the building is approximately **34.64 m**.