A T V transmitting antenna is 80 m tall. If the receiving antenna is on the ground The service area is

Options

A. `12pi` sq km
B. `144pi` sq km
C. `1024pi` sq km
D. `32pi` sq km

Answer

To find the service area of a TV transmitting antenna that is 80 meters tall, we can use the formula for the distance to the horizon, which is given by:

\[ d = \sqrt{2 \cdot R \cdot h} \]

where:
- \( d \) is the distance to the horizon (in meters),
- \( R \) is the radius of the Earth (approximately 6400 km or \( 6.4 \times 10^6 \) meters),
- \( h \) is the height of the antenna (in meters).

### Step 1: Calculate the distance to the horizon for the transmitting antenna.

Given:
- Height of the transmitting antenna, \( h_t = 80 \) m
- Radius of the Earth, \( R = 6400 \) km = \( 6.4 \times 10^6 \) m

Using the formula:

\[ d_t = \sqrt{2 \cdot R \cdot h_t} \]

Substituting the values:

\[ d_t = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 80} \]

Calculating:

\[ d_t = \sqrt{2 \cdot 6.4 \times 10^6 \cdot 80} = \sqrt{1.024 \times 10^9} \approx 32000 \text{ m} \]

### Step 2: Convert the distance from meters to kilometers.

Since \( 1 \text{ km} = 1000 \text{ m} \):

\[ d_t = \frac{32000 \text{ m}}{1000} = 32 \text{ km} \]

### Step 3: Calculate the total service area.

The service area \( A \) can be calculated using the formula for the area of a circle:

\[ A = \pi d^2 \]

Substituting \( d = 32 \) km:

\[ A = \pi (32)^2 \]

Calculating:

\[ A = \pi \cdot 1024 \approx 3215.99 \text{ km}^2 \]

### Final Result

Thus, the service area of the TV transmitting antenna is approximately:

\[ A \approx 3216 \text{ km}^2 \]