The height of a TV antenna is 200 m. The population density is `4000 km^(-2)` . Find the population benefited

Options

A. `3.2 xx 10^(8)`
B. `3.2 xx 10^(7)`
C. `3.2 xx 10^(6)`
D. `3.2 xx 10^(5)`

Answer

To solve the problem, we need to find the population that can benefit from a TV antenna that is 200 meters tall, given a population density of 4000 people per square kilometer. We will follow these steps:

### Step 1: Calculate the Distance Covered by the Antenna
The distance \( d \) covered by the TV antenna can be calculated using the formula:
\[ d = \sqrt{2 \cdot r \cdot h} \]
where:
- \( r \) is the radius of the Earth (approximately \( 6.4 \times 10^6 \) meters),
- \( h \) is the height of the antenna (200 meters).

Substituting the values:
\[ d = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 200} \]

### Step 2: Calculate the Area Covered by the Antenna
The area \( A \) covered by the antenna can be calculated using the formula:
\[ A = \pi \cdot d^2 \]

### Step 3: Convert Population Density to People per Square Meter
The population density is given as \( 4000 \, \text{people/km}^2 \). We need to convert this to people per square meter:
\[ 4000 \, \text{people/km}^2 = \frac{4000}{10^6} \, \text{people/m}^2 = 4 \times 10^{-3} \, \text{people/m}^2 \]

### Step 4: Calculate the Total Population Benefited
The total population that can benefit from the antenna is given by:
\[ \text{Total Population} = \text{Population Density} \times \text{Area} \]
Substituting the values we have:
\[ \text{Total Population} = (4 \times 10^{-3}) \cdot A \]

### Step 5: Substitute Area and Calculate Total Population
Now we substitute the area \( A \) calculated in Step 2 into the equation:
\[ \text{Total Population} = (4 \times 10^{-3}) \cdot \pi \cdot d^2 \]

### Step 6: Final Calculation
After calculating \( d \) and substituting it back into the equation, we can compute the total population.

After performing the calculations, we find:
\[ \text{Total Population} \approx 3.2 \times 10^7 \]

### Conclusion
Thus, the total population benefited by the TV antenna is approximately \( 3.2 \times 10^7 \).

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