The maximum distance between the transmitting and receiving TV towers is D. If the heights of both transmitting and receiving tower are doubled then the maximum distance between them becomes.

Options

A. `sqrt(2)D`
B. 2D
C. 4D
D. `D//2`

Answer

To solve the problem, we need to determine how the maximum distance \( D \) between the transmitting and receiving TV towers changes when the heights of both towers are doubled.

### Step-by-Step Solution:

1. **Identify the Initial Heights**:
Let the height of the transmitting tower be \( H_T \) and the height of the receiving tower be \( H_R \).

2. **Use the Formula for Maximum Distance**:
The maximum distance \( D \) between the transmitting and receiving towers is given by the formula:
\[ D = \sqrt{2 H_R R} + \sqrt{2 H_T R} \]
where \( R \) is the radius of the Earth.

3. **Double the Heights**:
When the heights of both towers are doubled, we have:
\[ H_T' = 2 H_T \quad \text{and} \quad H_R' = 2 H_R \]

4. **Substitute the New Heights into the Formula**:
Now, substituting the new heights into the distance formula gives:
\[ D' = \sqrt{2 H_R' R} + \sqrt{2 H_T' R} \]
Substituting \( H_R' \) and \( H_T' \):
\[ D' = \sqrt{2 (2 H_R) R} + \sqrt{2 (2 H_T) R} \]

5. **Simplify the Expression**:
This simplifies to:
\[ D' = \sqrt{4 H_R R} + \sqrt{4 H_T R} \]
\[ D' = 2 \sqrt{2 H_R R} + 2 \sqrt{2 H_T R} \]
\[ D' = 2 \left( \sqrt{2 H_R R} + \sqrt{2 H_T R} \right) \]
\[ D' = 2D \]

6. **Final Result**:
Therefore, the maximum distance between the towers when their heights are doubled is:
\[ D' = 2D \]

### Conclusion:
The maximum distance between the transmitting and receiving TV towers becomes \( 2D \) when the heights of both towers are doubled.

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