To double the covering range of a TV transimitter tower, its height should be made

Options

A. `(1)/(sqrt(2))`
B. 2
C. 4
D. `sqrt(2)`

Explanation

To solve the problem of how much to increase the height of a TV transmitter tower to double its coverage range, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Coverage Range**:
The coverage range \( d \) of a TV transmitter tower can be expressed as:
\[ d = \sqrt{2 R H_T} + \sqrt{2 R H_R} \]
where:
- \( R \) is the radius of the Earth,
- \( H_T \) is the height of the transmitter tower,
- \( H_R \) is the height of the receiver.

2. **Assume Receiver Height**:
In this scenario, we assume the receiver is at ground level, which means \( H_R = 0 \). Therefore, the equation simplifies to:
\[ d = \sqrt{2 R H_T} \]

3. **Set Up the Equation for Doubling the Range**:
We want to double the coverage range, so we set the new range \( d' = 2d \):
\[ d' = 2 \sqrt{2 R H_T} \]

4. **Express the New Range in Terms of Height**:
The new range can also be expressed as:
\[ d' = \sqrt{2 R H'_T} \]
where \( H'_T \) is the new height of the transmitter tower.

5. **Equate the Two Expressions for the New Range**:
Equating the two expressions for the new range gives:
\[ 2 \sqrt{2 R H_T} = \sqrt{2 R H'_T} \]

6. **Square Both Sides to Eliminate the Square Root**:
Squaring both sides results in:
\[ 4(2 R H_T) = 2 R H'_T \]

7. **Simplify the Equation**:
Dividing both sides by \( 2R \) (assuming \( R \neq 0 \)):
\[ 4 H_T = H'_T \]

8. **Conclusion**:
This means that the new height \( H'_T \) must be four times the original height \( H_T \):
\[ H'_T = 4 H_T \]

### Final Answer:
To double the coverage range of a TV transmitter tower, its height should be made **four times** the original height.

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