The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is `60^@` and the angle of elevation of the top iof the second tower from the foot of the first tower is `30^@`. The distance between the two towers isntimes the height of the shorter tower. What is nequal to?

Options

A. `sqrt 2`
B. `sqrt 3`
C. `1/2`
D. `1/3`

Answer

To solve the problem, we need to analyze the given information about the two towers and use trigonometric principles to find the value of \( n \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let Tower 1 (AB) be 30 m high.
- Let Tower 2 (CD) be of height \( h \).
- The angle of elevation from the foot of Tower 2 (point D) to the top of Tower 1 (point B) is \( 60^\circ \).
- The angle of elevation from the foot of Tower 1 (point A) to the top of Tower 2 (point C) is \( 30^\circ \).
- The distance between the two towers is \( n \) times the height of the shorter tower.

2. **Using Triangle ABD**:
- In triangle ABD, we have:
\[ \tan(60^\circ) = \frac{AB}{BD} \]
Here, \( AB = 30 \) m (height of Tower 1) and \( \tan(60^\circ) = \sqrt{3} \).
\[ \sqrt{3} = \frac{30}{BD} \]
Rearranging gives:
\[ BD = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ m} \]

3. **Using Triangle CDB**:
- In triangle CDB, we have:
\[ \tan(30^\circ) = \frac{CD}{BD} \]
Here, \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
\[ \frac{1}{\sqrt{3}} = \frac{h}{10\sqrt{3}} \]
Rearranging gives:
\[ h = \frac{10\sqrt{3}}{\sqrt{3}} = 10 \text{ m} \]

4. **Finding the Distance Between the Towers**:
- The distance between the two towers is given as \( n \) times the height of the shorter tower. Since the height of Tower 2 (CD) is \( 10 \) m, and Tower 1 (AB) is \( 30 \) m, the shorter tower is Tower 2.
- Therefore, the distance \( d \) between the two towers is:
\[ d = n \times 10 \]

5. **Using the Relationship**:
- From the triangle ABD, we found \( BD = 10\sqrt{3} \).
- Setting the two expressions for distance equal gives:
\[ n \times 10 = 10\sqrt{3} \]
- Dividing both sides by 10:
\[ n = \sqrt{3} \]

### Conclusion:
The value of \( n \) is \( \sqrt{3} \).