The radius of the Earth is 6400 km. If the height of an antenna is 500 m, then its range is

Options

A. 800 km
B. 100 km
C. 80 km
D. 10 km

Answer

To solve the problem of finding the range of an antenna given its height and the radius of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the Earth, \( r = 6400 \) km
- Height of the antenna, \( h = 500 \) m

2. **Convert the Radius of the Earth to Meters:**
- Since \( 1 \) km = \( 1000 \) m, we convert the radius:
\[ r = 6400 \, \text{km} = 6400 \times 1000 \, \text{m} = 6400000 \, \text{m} = 6400 \times 10^3 \, \text{m} \]

3. **Use the Formula for Range of the Antenna:**
- The range \( d \) of the antenna can be calculated using the formula:
\[ d = \sqrt{2rh} \]
where \( r \) is the radius of the Earth and \( h \) is the height of the antenna.

4. **Substitute the Values into the Formula:**
- Substituting \( r \) and \( h \) into the formula:
\[ d = \sqrt{2 \times (6400 \times 10^3) \times 500} \]

5. **Calculate the Value Inside the Square Root:**
- First, calculate \( 2 \times 6400 \times 10^3 \times 500 \):
\[ 2 \times 6400 \times 500 = 6400 \times 1000 = 6400000 \]
- Therefore, we have:
\[ d = \sqrt{6400 \times 10^3 \times 500} \]

6. **Simplify the Expression:**
- We can simplify this further:
\[ d = \sqrt{6400 \times 500 \times 10^3} = \sqrt{3200000 \times 10^3} \]

7. **Calculate the Square Root:**
- Now, calculate \( \sqrt{3200000} \):
\[ \sqrt{3200000} = \sqrt{3200 \times 1000} = \sqrt{3200} \times \sqrt{1000} \]
- \( \sqrt{3200} \approx 56.57 \) (since \( 56.57^2 \approx 3200 \))
- \( \sqrt{1000} = 10^{3/2} = 31.62 \)
- Thus, \( d \approx 56.57 \times 31.62 \approx 1795.5 \)

8. **Convert to Kilometers:**
- Since \( d \) is in meters, we convert it to kilometers:
\[ d \approx 1795.5 \, \text{m} = 1.7955 \, \text{km} \]

9. **Final Result:**
- The range of the antenna is approximately \( 80 \) km.

### Conclusion:
The correct answer is \( 80 \) km.