TV tower has a height 100m. What is the population density around the T.V Tower if  the total population covered is 50 lac? Radius of earth `6.4 x 10^(6)`m 

Answer

To find the population density around the TV tower, we can follow these steps:

### Step 1: Understand the Given Data
- Height of the TV tower (H) = 100 m
- Total population covered (P) = 50 lakh = 50 × 10^5 = 5,000,000
- Radius of the Earth (R) = 6.4 × 10^6 m

### Step 2: Calculate the Distance (d) Covered by the Tower
The distance (d) that the signal can cover from the tower can be calculated using the formula:
\[ d = \sqrt{2 \cdot H \cdot R} \]
Substituting the values:
\[ d = \sqrt{2 \cdot 100 \cdot 6.4 \times 10^6} \]

### Step 3: Calculate d
Calculating the value:
\[ d = \sqrt{2 \cdot 100 \cdot 6.4 \times 10^6} = \sqrt{1280000} \approx 1131.37 \, \text{m} \]

### Step 4: Calculate the Area (A) Covered
The area (A) covered by the tower can be calculated using the formula for the area of a circle:
\[ A = \pi d^2 \]
Substituting the value of d:
\[ A = \pi (1131.37)^2 \]
Calculating the area:
\[ A \approx 3.14 \times 1280000 \approx 4021232.8 \, \text{m}^2 \]

### Step 5: Calculate Population Density (ρ)
Population density (ρ) can be calculated using the formula:
\[ \rho = \frac{P}{A} \]
Substituting the values:
\[ \rho = \frac{5000000}{4021232.8} \]

### Step 6: Calculate ρ
Calculating the population density:
\[ \rho \approx 1.243 \, \text{people/m}^2 \]

### Step 7: Convert to Appropriate Units
To express the population density in terms of people per square kilometer, we convert:
\[ 1 \, \text{m}^2 = 10^{-6} \, \text{km}^2 \]
Thus,
\[ \rho \approx 1.243 \times 10^6 \, \text{people/km}^2 \approx 1242.9 \, \text{people/km}^2 \]

### Final Answer
The population density around the TV tower is approximately **1242.9 people/km²**.

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