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 TV telecast, we can use the formula:

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

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

### Step-by-Step Solution:

1. **Convert the distance from kilometers to meters**:
- Given the distance \( d = 128 \) km.
- To convert kilometers to meters, we multiply by \( 10^3 \):
\[ d = 128 \times 10^3 = 128000 \text{ meters} \]

2. **Know the radius of the Earth**:
- The radius of the Earth \( r \) is given as \( 6.4 \times 10^6 \) meters.

3. **Substitute the values into the formula**:
- Now we can substitute \( d \) and \( r \) into the formula:
\[ h = \frac{(128000)^2}{2 \times (6.4 \times 10^6)} \]

4. **Calculate \( d^2 \)**:
- First, calculate \( (128000)^2 \):
\[ (128000)^2 = 16384000000 \text{ square meters} \]

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

6. **Substitute back into the height formula**:
- Now substitute these values back into the height formula:
\[ h = \frac{16384000000}{12.8 \times 10^6} \]

7. **Perform the division**:
- Calculate the height:
\[ h = \frac{16384000000}{12800000} = 1280 \text{ meters} \]

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