If the range of single transmission between sending and reciving antennas of equal heights is 45 km. find height of antennas

Options

A. 30 m
B. 39.5 m
C. 45 m
D. 64 m

Answer

To find the height of the antennas given the range of transmission, we can use the formula for the range of radio waves between two antennas of equal height. The formula is:

\[ D = \sqrt{2hR} \]

where:
- \( D \) is the distance (range) between the antennas,
- \( h \) is the height of the antennas,
- \( R \) is the radius of the Earth (approximately \( 6.4 \times 10^6 \) meters).

### Step-by-Step Solution:

1. **Identify the given values:**
- The range \( D \) is given as 45 km. We need to convert this to meters:
\[ D = 45 \text{ km} = 45 \times 10^3 \text{ m} = 45000 \text{ m} \]

2. **Use the formula for range:**
Since both antennas are of equal height, we can express the range as:
\[ D = 2 \sqrt{2hR} \]
This is because the range is the sum of the distances from each antenna to the horizon.

3. **Substituting the values into the formula:**
We can rearrange the formula to solve for height \( h \):
\[ 45000 = 2 \sqrt{2h \times 6.4 \times 10^6} \]

4. **Isolate the square root:**
Divide both sides by 2:
\[ 22500 = \sqrt{2h \times 6.4 \times 10^6} \]

5. **Square both sides to eliminate the square root:**
\[ 22500^2 = 2h \times 6.4 \times 10^6 \]
Calculate \( 22500^2 \):
\[ 22500^2 = 506250000 \]

6. **Rearranging to find \( h \):**
\[ 506250000 = 2h \times 6.4 \times 10^6 \]
\[ h = \frac{506250000}{2 \times 6.4 \times 10^6} \]

7. **Calculate the denominator:**
\[ 2 \times 6.4 \times 10^6 = 12.8 \times 10^6 \]

8. **Final calculation for height \( h \):**
\[ h = \frac{506250000}{12.8 \times 10^6} = \frac{506250000}{12800000} \approx 39.55 \text{ m} \]

### Conclusion:
The height of each antenna is approximately **39.55 meters**.