The height of a TV transmitting antenna is 20m. The telecast can cover a radius of

Options

A. 8km
B. 16km
C. 32km
D. 64 km

Answer

To find the radius of coverage of a TV transmitting antenna given its height, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Height of the Antenna**:
The height of the TV transmitting antenna (H) is given as 20 meters.

2. **Know the Radius of the Earth**:
The average radius of the Earth (R) is approximately 6400 kilometers. We need to convert this into meters for consistency:
\[ R = 6400 \, \text{km} = 6400 \times 1000 \, \text{m} = 6.4 \times 10^6 \, \text{m} \]

3. **Use the Formula for Distance**:
The formula to calculate the distance (d) covered by the antenna based on its height is:
\[ d = \sqrt{2 \cdot R \cdot H} \]
where:
- \(d\) is the distance (radius of coverage),
- \(R\) is the radius of the Earth,
- \(H\) is the height of the antenna.

4. **Substitute the Values**:
Now, we substitute the known values into the formula:
\[ d = \sqrt{2 \cdot (6.4 \times 10^6 \, \text{m}) \cdot (20 \, \text{m})} \]

5. **Calculate Inside the Square Root**:
First, calculate the product inside the square root:
\[ 2 \cdot (6.4 \times 10^6) \cdot 20 = 256000000 \, \text{m}^2 \]

6. **Take the Square Root**:
Now, take the square root of the result:
\[ d = \sqrt{256000000} \approx 16000 \, \text{m} \]

7. **Convert to Kilometers**:
Finally, convert the distance from meters to kilometers:
\[ d = \frac{16000 \, \text{m}}{1000} = 16 \, \text{km} \]

### Final Answer:
The radius of coverage of the TV transmitting antenna is **16 kilometers**.

---