The angle of elevation of the top of a tower of height H from the foot of another tower in the same plane is `60^(@)` and the angle of elevation of the top of the second tower from the foot of the first tower is `30^(@)`. If h is the height of the other tower, then which one of the following is correct?

Options

A. H=2h
B. `H=sqrt(3h)`
C. H=3h
D. None of the above

Explanation

To solve the problem, we will analyze the situation using trigonometric principles, specifically focusing on the right triangles formed by the two towers and the angles of elevation.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the first tower be \( H \).
- Let the height of the second tower be \( h \).
- The angle of elevation from the foot of the first tower to the top of the second tower is \( 30^\circ \).
- The angle of elevation from the foot of the second tower to the top of the first tower is \( 60^\circ \).

2. **Setting Up the Triangles**:
- From the foot of the first tower (let's call it point A) to the top of the second tower (point B), we have a right triangle \( AOB \) where:
- \( \angle AOB = 30^\circ \)
- The opposite side (height of the second tower) is \( h \).
- The adjacent side (distance between the two towers) is \( x \).
- From the foot of the second tower (point C) to the top of the first tower (point D), we have another right triangle \( COD \) where:
- \( \angle COD = 60^\circ \)
- The opposite side (height of the first tower) is \( H \).
- The adjacent side (same distance \( x \)) is the same.

3. **Using Trigonometric Ratios**:
- For triangle \( AOB \) (using \( \tan \)):
\[ \tan(30^\circ) = \frac{h}{x} \]
We know \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies x = h \sqrt{3} \]

- For triangle \( COD \) (using \( \tan \)):
\[ \tan(60^\circ) = \frac{H}{x} \]
We know \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{H}{x} \implies x = \frac{H}{\sqrt{3}} \]

4. **Equating the Two Expressions for \( x \)**:
- From the two equations we derived for \( x \):
\[ h \sqrt{3} = \frac{H}{\sqrt{3}} \]

5. **Solving for \( H \)**:
- Cross-multiplying gives:
\[ h \sqrt{3} \cdot \sqrt{3} = H \implies 3h = H \]

6. **Final Relation**:
- Therefore, the relationship between the heights of the two towers is:
\[ H = 3h \]

### Conclusion:
The correct relationship between the heights of the two towers is \( H = 3h \).