A TV transmission tower antenna is at a height of 20m. The percentage increase in area covered in case if the receiving antenna is at ground level to that at a height of 25 m is (Radius of earth `=6.4xx10^(6)m`)

Options

A. 2.48
B. `348.9%`
C. `150%`
D. `360.2%`

Explanation

To solve the problem of finding the percentage increase in area covered by a TV transmission tower when the receiving antenna is at ground level compared to when it is at a height of 25 m, we can follow these steps:

### Step 1: Calculate the distance covered by the transmission tower antenna (Dt)
The formula to calculate the distance (Dt) covered by the transmission tower is given by:
\[ D_t = \sqrt{2 h_t R_e} \]
where:
- \( h_t = 20 \, \text{m} \) (height of the transmission tower)
- \( R_e = 6.4 \times 10^6 \, \text{m} \) (radius of the Earth)

Substituting the values:
\[ D_t = \sqrt{2 \times 20 \times 6.4 \times 10^6} \]
Calculating this gives:
\[ D_t = \sqrt{2560000} \approx 1600 \, \text{m} = 16 \, \text{km} \]

### Step 2: Calculate the area covered by the transmission tower antenna (At)
The area covered by the transmission tower can be calculated using the formula for the area of a circle:
\[ A_t = \pi D_t^2 \]
Substituting the value of \( D_t \):
\[ A_t = \pi (16 \times 10^3)^2 \]
Calculating this gives:
\[ A_t \approx 3.14 \times 256 \times 10^6 \approx 803.84 \, \text{km}^2 \]

### Step 3: Calculate the maximum distance covered by the receiving tower antenna (Dm)
The formula for the maximum distance (Dm) when the receiving antenna is at a height \( h_r \) is:
\[ D_m = \sqrt{2 h_t R_e} + \sqrt{2 h_r R_e} \]
where \( h_r = 25 \, \text{m} \).

Substituting the values:
\[ D_m = \sqrt{2 \times 20 \times 6.4 \times 10^6} + \sqrt{2 \times 25 \times 6.4 \times 10^6} \]
Calculating each term:
- First term: \( D_t \approx 16 \, \text{km} \)
- Second term:
\[ \sqrt{2 \times 25 \times 6.4 \times 10^6} = \sqrt{3200000} \approx 1788.85 \, \text{m} \approx 1.79 \, \text{km} \]
Thus:
\[ D_m \approx 16 + 1.79 \approx 33.79 \, \text{km} \]

### Step 4: Calculate the area covered by the receiving tower antenna (Ar)
Using the area formula:
\[ A_r = \pi D_m^2 \]
Substituting the value of \( D_m \):
\[ A_r = \pi (33.79 \times 10^3)^2 \]
Calculating this gives:
\[ A_r \approx 3.14 \times 1145.84 \times 10^6 \approx 3608.52 \, \text{km}^2 \]

### Step 5: Calculate the percentage increase in area
The percentage increase in area is given by:
\[ \text{Percentage Increase} = \frac{A_r - A_t}{A_t} \times 100 \]
Substituting the areas:
\[ \text{Percentage Increase} = \frac{3608.52 - 803.84}{803.84} \times 100 \]
Calculating this gives:
\[ \text{Percentage Increase} \approx \frac{2804.68}{803.84} \times 100 \approx 348.9\% \]

### Final Answer:
The percentage increase in area covered when the receiving antenna is at ground level compared to when it is at a height of 25 m is approximately **348.9%**.