A TV transmitting antenna is 98 m high and the receiving antenna is at the ground level . If the radius of the earth is `6400 km` ,the surface area covered by the transmitting antenna is approximately:

Options

A. `1549 km^(2)`
B. `1240 km^(2)`
C. `3942 km^(2)`
D. `4868 km ^(2)`

Answer

To solve the problem of finding the surface area covered by a TV transmitting antenna that is 98 m high, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a TV transmitting antenna that is 98 m high and a receiving antenna at ground level. The radius of the Earth is given as 6400 km. We need to find the surface area covered by the transmitting antenna.

2. **Use the Formula for Distance**:
The distance \(d\) to the horizon from a height \(h\) is given by the formula:
\[ d = \sqrt{2rh} \]
where \(r\) is the radius of the Earth and \(h\) is the height of the antenna.

3. **Convert Units**:
First, we need to convert the radius of the Earth from kilometers to meters:
\[ r = 6400 \text{ km} = 6400 \times 10^3 \text{ m} = 6.4 \times 10^6 \text{ m} \]

4. **Calculate the Distance**:
Substitute the values of \(r\) and \(h\) into the formula:
\[ d = \sqrt{2 \times (6.4 \times 10^6) \times 98} \]
Calculate \(2 \times 6.4 \times 10^6 \times 98\):
\[ 2 \times 6.4 \times 10^6 \times 98 = 1.25472 \times 10^9 \]
Now take the square root:
\[ d = \sqrt{1.25472 \times 10^9} \approx 35425 \text{ m} \]

5. **Calculate the Surface Area**:
The surface area \(A\) covered by the antenna is given by:
\[ A = \pi d^2 \]
Substitute \(d\) into the formula:
\[ A = \pi (35425)^2 \]
Calculate \(35425^2\):
\[ 35425^2 \approx 1254875625 \]
Now calculate the area:
\[ A \approx \pi \times 1254875625 \approx 3.941 \times 10^9 \text{ m}^2 \]

6. **Convert to Square Kilometers**:
Since \(1 \text{ km}^2 = 10^6 \text{ m}^2\), we convert the area:
\[ A \approx \frac{3.941 \times 10^9}{10^6} \approx 3941 \text{ km}^2 \]

7. **Final Result**:
The surface area covered by the transmitting antenna is approximately:
\[ A \approx 3941 \text{ km}^2 \]