A TV tower has a height of 100m. How much population is covered by TV broadcast. If the average population density around the tower is `1000 km^(-2)` ? (radius of earth =`6.4xx10^(6)m`)

Options

A. `6xx10^(6)`
B. `2xx10^(6)`
C. `12xx10^(6)`
D. `4xx10^(6)`

Answer

To solve the problem of determining how much population is covered by the TV broadcast from a tower of height 100 m, we can follow these steps:

### Step 1: Calculate the maximum radius covered by the TV broadcast
The maximum radius \( d \) that can be covered by the TV broadcast is given by the formula:
\[ d = \sqrt{2rh} \]
where:
- \( r \) is the radius of the Earth (given as \( 6.4 \times 10^6 \) m),
- \( h \) is the height of the TV tower (given as 100 m).

Substituting the values:
\[ d = \sqrt{2 \times (6.4 \times 10^6) \times 100} \]

### Step 2: Calculate the value of \( d \)
Calculating inside the square root:
\[ d = \sqrt{2 \times 6.4 \times 10^6 \times 100} = \sqrt{1.28 \times 10^9} = 3.58 \times 10^4 \text{ m} \quad (\text{approximately}) \]

### Step 3: Calculate the area covered by the broadcast
The area \( A \) covered by the broadcast can be calculated using the formula for the area of a circle:
\[ A = \pi d^2 \]
Substituting the value of \( d \):
\[ A = \pi (3.58 \times 10^4)^2 \]

### Step 4: Calculate the area
Calculating \( d^2 \):
\[ d^2 = (3.58 \times 10^4)^2 = 1.28 \times 10^9 \text{ m}^2 \]
Now substituting into the area formula:
\[ A = \pi \times 1.28 \times 10^9 \approx 4.02 \times 10^9 \text{ m}^2 \quad (\text{using } \pi \approx 3.14) \]

### Step 5: Calculate the total population covered
The total population \( P \) covered by the broadcast can be calculated using the formula:
\[ P = A \times P_{\text{average}} \]
where \( P_{\text{average}} \) is the average population density (given as \( 1000 \text{ people/km}^2 \)). First, we need to convert this density into \( \text{people/m}^2 \):
\[ P_{\text{average}} = 1000 \text{ people/km}^2 = \frac{1000}{10^6} \text{ people/m}^2 = 1 \times 10^{-3} \text{ people/m}^2 \]

Now substituting the values:
\[ P = 4.02 \times 10^9 \text{ m}^2 \times 1 \times 10^{-3} \text{ people/m}^2 \]
\[ P \approx 4.02 \times 10^6 \text{ people} \]

### Final Answer
The total population covered by the TV broadcast is approximately \( 4.02 \times 10^6 \) people.