The height of a television tower is 100 m . If radius of earth is `6.4 xx 10^6` and average- population density surrounding the tower is 1000 per ` km^(2)` , then the population covered by the television transmission is

Options

A. `2.06 xx 10^6`
B. `4.02 xx 10^6`
C. `5.18 xx 10^9`
D. `6.04 xx 10^9`

Explanation

To solve the problem, we need to find the population covered by the television transmission from a tower of height 100 m, given the radius of the Earth and the population density. Here’s a step-by-step solution:

### Step 1: Calculate the range covered by the television tower
The range (D) covered by the tower can be calculated using the formula:
\[ D = \sqrt{2 \cdot h \cdot R} \]
where:
- \( h \) is the height of the tower (100 m),
- \( R \) is the radius of the Earth (6.4 × 10^6 m).

### Calculation:
Substituting the values:
\[ D = \sqrt{2 \cdot 100 \, \text{m} \cdot 6.4 \times 10^6 \, \text{m}} \]
\[ D = \sqrt{1280000} \]
\[ D \approx 1131.37 \, \text{m} \]

### Step 2: Calculate the area covered
The area (A) covered by the transmission can be calculated using the formula for the area of a circle:
\[ A = \pi D^2 \]

### Calculation:
Substituting the value of D:
\[ A = \pi (1131.37 \, \text{m})^2 \]
\[ A \approx 3.14 \times 1280000 \]
\[ A \approx 4021230.68 \, \text{m}^2 \]

### Step 3: Convert the area to km²
Since the population density is given in km², we need to convert the area from m² to km²:
\[ 1 \, \text{km}^2 = 10^6 \, \text{m}^2 \]
\[ A \approx \frac{4021230.68 \, \text{m}^2}{10^6} \]
\[ A \approx 4.02 \, \text{km}^2 \]

### Step 4: Calculate the population covered
Now, we can calculate the population covered using the population density:
\[ \text{Population} = \text{Population Density} \times \text{Area} \]
Given that the population density is 1000 per km²:
\[ \text{Population} = 1000 \, \text{people/km}^2 \times 4.02 \, \text{km}^2 \]
\[ \text{Population} \approx 4020 \, \text{people} \]

### Final Answer:
The population covered by the television transmission is approximately **4020 people**.

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