The angle of elevation of the top of two vertical towers as seen from the middle point of the line joining the foot of the towers are `60^(@ )and 30^(@)` respectively. The ratio of the heights of the tower is

Options

A. `2:1`
B. `sqrt(3):1`
C. `3:2`
D. `3:1`

Answer

To solve the problem of finding the ratio of the heights of two vertical towers based on the angles of elevation observed from the midpoint between them, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Setup**:
- Let the heights of the two towers be \( h_1 \) and \( h_2 \).
- The angles of elevation from the midpoint to the tops of the towers are \( 60^\circ \) and \( 30^\circ \) respectively.

2. **Defining the Distances**:
- Let the distance from the midpoint to the foot of each tower be \( x \). Therefore, the total distance between the two towers is \( 2x \).

3. **Using Trigonometry for the First Tower**:
- For the first tower (height \( h_1 \)) with an angle of elevation of \( 60^\circ \):
\[ \tan(60^\circ) = \frac{h_1}{x} \]
- We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h_1}{x} \]
- Rearranging gives:
\[ h_1 = x \sqrt{3} \]

4. **Using Trigonometry for the Second Tower**:
- For the second tower (height \( h_2 \)) with an angle of elevation of \( 30^\circ \):
\[ \tan(30^\circ) = \frac{h_2}{x} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h_2}{x} \]
- Rearranging gives:
\[ h_2 = \frac{x}{\sqrt{3}} \]

5. **Finding the Ratio of Heights**:
- Now, we can find the ratio of the heights \( h_1 \) and \( h_2 \):
\[ \frac{h_1}{h_2} = \frac{x \sqrt{3}}{\frac{x}{\sqrt{3}}} \]
- Simplifying this gives:
\[ \frac{h_1}{h_2} = \frac{x \sqrt{3} \cdot \sqrt{3}}{x} = \frac{3x}{x} = 3 \]
- Thus, the ratio of the heights is:
\[ h_1 : h_2 = 3 : 1 \]

### Final Answer
The ratio of the heights of the two towers is \( 3 : 1 \).