A T.V. transmitting antenna is 128 m tall . If the receiving antenna is at the ground level , the maximum distance between them for satisfactory communication in L.O.S. mode is
(Radius of the earth = `6.4 xx 10^(6)` m )

Options

A. `64 xx sqrt(10)` km
B. `(128)/(sqrt(10))` km
C. `128 xx sqrt(10)` km
D. `(64)/(sqrt(10))` km

Answer

To solve the problem, we need to find the maximum distance for satisfactory communication between a TV transmitting antenna and a receiving antenna at ground level using the Line of Sight (L.O.S) mode. The height of the transmitting antenna is given as 128 m, and the radius of the Earth is \(6.4 \times 10^6\) m.

### Step-by-Step Solution:

1. **Identify the formula for maximum distance in L.O.S communication:**
The formula to calculate the maximum distance \(d\) between the transmitting antenna and the receiving antenna is given by:
\[ d = \sqrt{2 h_t R} \]
where:
- \(h_t\) = height of the transmitting antenna (in meters)
- \(R\) = radius of the Earth (in meters)

2. **Substitute the known values into the formula:**
Here, \(h_t = 128\) m and \(R = 6.4 \times 10^6\) m. Plugging these values into the formula, we get:
\[ d = \sqrt{2 \times 128 \times 6.4 \times 10^6} \]

3. **Calculate the product inside the square root:**
First, calculate \(2 \times 128\):
\[ 2 \times 128 = 256 \]
Now, substitute this back into the equation:
\[ d = \sqrt{256 \times 6.4 \times 10^6} \]

4. **Simplify the expression:**
We can break this down further:
\[ d = \sqrt{256} \times \sqrt{6.4} \times \sqrt{10^6} \]
We know:
- \(\sqrt{256} = 16\)
- \(\sqrt{10^6} = 10^3 = 1000\)

Now we need to calculate \(\sqrt{6.4}\):
\[ \sqrt{6.4} \approx 2.5298 \]

5. **Combine the results:**
Now we can calculate \(d\):
\[ d \approx 16 \times 2.5298 \times 1000 \]
\[ d \approx 40.4768 \times 1000 \approx 40476.8 \text{ meters} \]

6. **Convert to kilometers:**
To convert meters to kilometers, divide by 1000:
\[ d \approx 40.4768 \text{ km} \]

### Final Answer:
The maximum distance between the transmitting antenna and the receiving antenna for satisfactory communication in L.O.S mode is approximately **40.48 km**.