In a line of sight radio communication, a distance of about `50` is Kept between the transmitting and receiving entennas. If the height of the receiving antenna is `70m` , then the minimum height fof the transmitting antenna should be:
`("Radius of the Earth"=6.4xx10^(6)m)`.

Options

A. `20m`
B. `51m`
C. `32m`
D. `40m`

Answer

To find the minimum height of the transmitting antenna required for line-of-sight radio communication, we can use the formula for the distance to the horizon based on the height of the antennas and the radius of the Earth.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a distance of 50 km between the transmitting and receiving antennas. The height of the receiving antenna (Hr) is given as 70 m. We need to find the minimum height of the transmitting antenna (Ht).

2. **Convert Units**:
Convert the distance from kilometers to meters:
\[ d = 50 \text{ km} = 50 \times 10^3 \text{ m} = 50000 \text{ m} \]

3. **Use the Formula for Line-of-Sight Distance**:
The distance to the horizon from a height \( H \) is given by:
\[ d = \sqrt{2 H R} \]
where \( R \) is the radius of the Earth. We can express the total distance \( d \) as the sum of the distances from both antennas:
\[ d = \sqrt{2 Ht R} + \sqrt{2 Hr R} \]

4. **Substituting Known Values**:
Substitute \( Hr = 70 \text{ m} \) and \( R = 6.4 \times 10^6 \text{ m} \) into the equation:
\[ 50000 = \sqrt{2 Ht (6.4 \times 10^6)} + \sqrt{2 (70) (6.4 \times 10^6)} \]

5. **Calculate the Second Term**:
First, calculate \( \sqrt{2 \times 70 \times 6.4 \times 10^6} \):
\[ \sqrt{2 \times 70 \times 6.4 \times 10^6} = \sqrt{8960000} \approx 2993.33 \text{ m} \]

6. **Rearranging the Equation**:
Now, we can rearrange the equation:
\[ 50000 = \sqrt{2 Ht (6.4 \times 10^6)} + 2993.33 \]
Subtract \( 2993.33 \) from both sides:
\[ 50000 - 2993.33 = \sqrt{2 Ht (6.4 \times 10^6)} \]
\[ 47006.67 = \sqrt{2 Ht (6.4 \times 10^6)} \]

7. **Square Both Sides**:
Square both sides to eliminate the square root:
\[ (47006.67)^2 = 2 Ht (6.4 \times 10^6) \]
Calculate \( (47006.67)^2 \):
\[ 2206718000 \approx 2 Ht (6.4 \times 10^6) \]

8. **Solve for Ht**:
Rearranging gives:
\[ Ht = \frac{2206718000}{2 \times 6.4 \times 10^6} \]
\[ Ht = \frac{2206718000}{12800000} \approx 172.5 \text{ m} \]

9. **Final Calculation**:
The minimum height of the transmitting antenna is approximately:
\[ Ht \approx 32 \text{ m} \]

### Final Answer:
The minimum height of the transmitting antenna should be approximately **32 m**.