Looking from the top of a 20 m high building, the angle of elevation of the top of a tower is `60^(@)` and the angle of depression of to its bottom is `30^(@).` What is he height of the tower?

Options

A. 50m
B. 60 m
C. 70m
D. 80m

Answer

To solve the problem of finding the height of the tower, we can break it down into several steps.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a building that is 20 meters high. From the top of this building, the angle of elevation to the top of a tower is \(60^\circ\), and the angle of depression to the bottom of the tower is \(30^\circ\). We need to find the height of the tower.

2. **Draw a Diagram**:
Draw a vertical line representing the building (AB) of height 20 m. Above this, draw a line representing the tower (CD). Mark points:
- A: Top of the building
- B: Bottom of the building
- C: Top of the tower
- D: Bottom of the tower

3. **Identify Angles**:
- The angle of elevation from A to C is \(60^\circ\).
- The angle of depression from A to D is \(30^\circ\).

4. **Use Trigonometry for the Angle of Depression**:
For triangle ABD:
- The height of the building (AB) = 20 m
- Let AE be the horizontal distance from the building to the tower.
- Using the angle of depression \(30^\circ\):
\[ \tan(30^\circ) = \frac{AB}{AE} \]
\[ \tan(30^\circ) = \frac{20}{AE} \]
- Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{20}{AE} \]
- Rearranging gives:
\[ AE = 20\sqrt{3} \]

5. **Use Trigonometry for the Angle of Elevation**:
For triangle AEC:
- Let EC be the height of the tower (CD).
- Using the angle of elevation \(60^\circ\):
\[ \tan(60^\circ) = \frac{EC}{AE} \]
\[ \tan(60^\circ) = \frac{EC}{20\sqrt{3}} \]
- Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{EC}{20\sqrt{3}} \]
- Rearranging gives:
\[ EC = 20\sqrt{3} \cdot \sqrt{3} = 60 \]

6. **Calculate the Total Height of the Tower**:
The total height of the tower (CD) is the sum of the height of the building (AB) and the height of the tower (EC):
\[ \text{Height of the tower} = AB + EC = 20 + 60 = 80 \text{ m} \]

### Final Answer:
The height of the tower is **80 meters**.

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