A TV transmitting antenna is 81 m tall. How much service area it can cover if the receiving antenna is at the ground level? (Radius of earth=`6.4xx10^(6)m`)

Options

A. `3258km^(2)`
B. `4180km^(2)`
C. `2510km^(2)`
D. `1525km^(2)`

Explanation

To solve the problem of how much service area a TV transmitting antenna can cover, we can use the formula for the area covered by a transmitting antenna. The formula is given by:

\[ A = \pi D^2 \]

where \( D \) is the distance to the horizon, which can be calculated using the formula:

\[ D = \sqrt{2hR} \]

Here:
- \( h \) is the height of the transmitting antenna,
- \( R \) is the radius of the Earth.

Given:
- Height of the antenna, \( h = 81 \) m,
- Radius of the Earth, \( R = 6.4 \times 10^6 \) m.

### Step 1: Calculate the distance to the horizon (D)

Using the formula:

\[ D = \sqrt{2hR} \]

Substituting the values:

\[ D = \sqrt{2 \times 81 \times 6.4 \times 10^6} \]

Calculating the product inside the square root:

\[ D = \sqrt{2 \times 81 \times 6.4 \times 10^6} = \sqrt{1.0368 \times 10^9} \]

Now, calculating the square root:

\[ D \approx 32160 \text{ m} \]

### Step 2: Calculate the service area (A)

Now, we can find the area covered using the formula:

\[ A = \pi D^2 \]

Substituting the value of \( D \):

\[ A = \pi (32160)^2 \]

Calculating \( (32160)^2 \):

\[ (32160)^2 = 103,243,0560 \]

Now, substituting this back into the area formula:

\[ A \approx \pi \times 103,243,0560 \]

Using \( \pi \approx \frac{22}{7} \):

\[ A \approx \frac{22}{7} \times 103,243,0560 \]

Calculating this gives:

\[ A \approx 3,258,000,000 \text{ m}^2 \]

### Step 3: Convert the area to km²

Since \( 1 \text{ km}^2 = 10^6 \text{ m}^2 \):

\[ A \approx \frac{3,258,000,000}{10^6} \text{ km}^2 \]

This simplifies to:

\[ A \approx 3258 \text{ km}^2 \]

### Final Answer

The service area that the TV transmitting antenna can cover is approximately \( 3258 \text{ km}^2 \).

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