A TV tower has a height of 100 m. The population density around the TV if the population covered is 60.288 lac, is

Options

A. `5 xx 10^(3)km^(-2)`
B. `1.5 xx 10^(3) km^(-2)`
C. `7.5 xx 10^(3) km^(-2)`
D. `10^(4) km^(-2)`

Answer

To find the population density around a TV tower with a height of 100 m and a population covered of 60.288 lakh, we can follow these steps:

### Step-by-Step Solution:

1. **Convert Height to Kilometers**:
- The height of the TV tower is given as 100 m.
- To convert meters to kilometers, we use the conversion factor \(1 \text{ km} = 1000 \text{ m}\).
\[ \text{Height in km} = \frac{100 \text{ m}}{1000} = 0.1 \text{ km} \]

2. **Convert Population to Standard Form**:
- The population covered is given as 60.288 lakh.
- 1 lakh = \(10^5\), so:
\[ \text{Population covered} = 60.288 \times 10^5 \]

3. **Calculate Diameter**:
- The formula for the diameter \(d\) of the area covered by the signal from the tower is:
\[ d = 2 \sqrt{R \cdot h} \]
where \(R\) is the radius of the Earth (approximately 6400 km) and \(h\) is the height of the tower in km.
\[ d = 2 \sqrt{6400 \text{ km} \times 0.1 \text{ km}} = 2 \sqrt{640} \approx 50.4 \text{ km} \]

4. **Calculate Area**:
- The area \(A\) covered can be calculated using the formula for the area of a circle:
\[ A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(25.2\right)^2 \approx 1992.96 \text{ km}^2 \]

5. **Calculate Population Density**:
- We know that population covered \(PC\) is equal to area \(A\) multiplied by population density \(\rho\):
\[ PC = A \cdot \rho \]
Rearranging gives:
\[ \rho = \frac{PC}{A} \]
Substituting the values:
\[ \rho = \frac{60.288 \times 10^5}{1992.96} \approx 3020.3 \text{ people/km}^2 \]

6. **Final Result**:
- The population density around the TV tower is approximately \(3020.3 \text{ people/km}^2\).