What should be the height of transmitting antenna if the TV telecast is to cover a radius of . 128 km?

Answer

To find the height of the transmitting antenna required to cover a radius of 128 km for a TV telecast, we can use the formula:

\[ h = \frac{d^2}{2r} \]

where:
- \( h \) is the height of the antenna,
- \( d \) is the distance (radius of coverage),
- \( r \) is the radius of the Earth.

### Step-by-Step Solution:

1. **Convert the radius from kilometers to meters**:
\[ d = 128 \text{ km} = 128 \times 10^3 \text{ m} = 128000 \text{ m} \]

2. **Know the radius of the Earth**:
\[ r = 6400 \text{ km} = 6400 \times 10^3 \text{ m} = 6.4 \times 10^6 \text{ m} \]

3. **Substitute the values into the formula**:
\[ h = \frac{(128000)^2}{2 \times (6.4 \times 10^6)} \]

4. **Calculate \( d^2 \)**:
\[ d^2 = (128000)^2 = 128^2 \times (10^3)^2 = 16384 \times 10^6 = 1.6384 \times 10^{10} \text{ m}^2 \]

5. **Calculate \( 2r \)**:
\[ 2r = 2 \times (6.4 \times 10^6) = 12.8 \times 10^6 \text{ m} \]

6. **Substitute \( d^2 \) and \( 2r \) back into the height formula**:
\[ h = \frac{1.6384 \times 10^{10}}{12.8 \times 10^6} \]

7. **Perform the division**:
\[ h = \frac{1.6384}{12.8} \times 10^{10 - 6} = \frac{1.6384}{12.8} \times 10^4 \]

8. **Calculate \( \frac{1.6384}{12.8} \)**:
\[ \frac{1.6384}{12.8} = 0.128 \]

9. **Final calculation**:
\[ h = 0.128 \times 10^4 = 1280 \text{ m} \]

### Conclusion:
The height of the transmitting antenna should be **1280 meters**.