The angle of depression of 47 m high building form the top of a tower 137 m high is `30^(@)` . Calculate the distance between the building and the tower .

Answer

To solve the problem, we need to find the distance between the building and the tower using the given information about their heights and the angle of depression.

### Step-by-Step Solution:

1. **Identify the Heights**:
- Height of the tower (AB) = 137 m
- Height of the building (CD) = 47 m

2. **Calculate the Height Difference**:
- The height difference (AE) between the tower and the building can be calculated as:
\[ AE = AB - CD = 137 \, \text{m} - 47 \, \text{m} = 90 \, \text{m} \]

3. **Understanding the Angle of Depression**:
- The angle of depression from the top of the tower (point A) to the top of the building (point C) is given as \(30^\circ\).
- This means that the angle of elevation from point C to point A is also \(30^\circ\) due to alternate interior angles.

4. **Set Up the Right Triangle**:
- In triangle ACD, we have:
- AE = 90 m (vertical height difference)
- DE = x (horizontal distance between the tower and the building)
- Angle ACD = \(30^\circ\)

5. **Use the Tangent Function**:
- From the right triangle, we can use the tangent of angle ACD:
\[ \tan(30^\circ) = \frac{AE}{DE} \]
- Substituting the known values:
\[ \tan(30^\circ) = \frac{90}{x} \]
- We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\).

6. **Set Up the Equation**:
- Now we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{90}{x} \]

7. **Cross-Multiply to Solve for x**:
- Cross-multiplying gives:
\[ x = 90 \sqrt{3} \]

8. **Final Calculation**:
- To find the numerical value of \(x\):
\[ x \approx 90 \times 1.732 \approx 155.88 \, \text{m} \]

### Conclusion:
The distance between the building and the tower is approximately \(90 \sqrt{3}\) meters or about \(155.88\) meters.