If the height of transmitting and recelving antennas are `80m` each,the maximum line of sight distance will be : Given : earth's radius `=6.4 times 10^(6) m`

Options

A. `36km`
B. `32km`
C. `28km`
D. `64km`

Answer

To solve the problem of finding the maximum line of sight distance between two antennas of equal height, we can use the formula for the maximum distance \( d_{\text{max}} \):

\[ d_{\text{max}} = \sqrt{2 \cdot r \cdot h} \]

where:
- \( r \) is the radius of the Earth,
- \( h \) is the height of the antennas.

Given:
- Height of each antenna, \( h = 80 \, \text{m} \)
- Radius of the Earth, \( r = 6.4 \times 10^6 \, \text{m} \)

### Step-by-step Solution:

1. **Identify the formula**: We will use the formula for the maximum line of sight distance, which is given by:
\[ d_{\text{max}} = \sqrt{2 \cdot r \cdot h} \]

2. **Substitute the values into the formula**:
\[ d_{\text{max}} = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 80} \]

3. **Calculate the product inside the square root**:
- First, calculate \( 2 \cdot 6.4 \):
\[ 2 \cdot 6.4 = 12.8 \]
- Next, multiply by 80:
\[ 12.8 \cdot 80 = 1024 \]

4. **Now substitute back into the equation**:
\[ d_{\text{max}} = \sqrt{1024 \times 10^6} \]

5. **Simplify the square root**:
- The square root of 1024 is 32:
\[ \sqrt{1024} = 32 \]
- The square root of \( 10^6 \) is \( 10^3 = 1000 \):
\[ \sqrt{10^6} = 10^3 \]
- Therefore:
\[ d_{\text{max}} = 32 \times 10^3 = 32000 \, \text{m} \]

6. **Convert meters to kilometers**:
\[ d_{\text{max}} = 32000 \, \text{m} = 32 \, \text{km} \]

### Final Answer:
The maximum line of sight distance is \( 32 \, \text{km} \).