A TV transmission tower of antenna is at a height of 20 m. How much service area can it cover if the receiving antenna is (i) at ground level (ii) at a height of 25 m ? Calculate the percentage increase in area covered in case (ii) relative to case (i).

Options

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

Explanation

To solve the problem, we need to calculate the service area covered by a TV transmission tower at two different heights of the receiving antenna: first at ground level and then at a height of 25 m. We will also calculate the percentage increase in the area covered when the antenna is at 25 m compared to when it is at ground level.

### Step-by-Step Solution:

1. **Identify the given data:**
- Height of the transmission tower (h_t) = 20 m
- Height of the receiving antenna (h_r) for case (i) = 0 m (ground level)
- Height of the receiving antenna (h_r) for case (ii) = 25 m
- Radius of the Earth (R) = 6.4 × 10^6 m

2. **Calculate the range (D) for case (i) where the receiving antenna is at ground level:**
- The formula for the range is given by:
\[ D = \sqrt{2 h_t R} \]
- Substituting the values:
\[ D = \sqrt{2 \times 20 \times 6.4 \times 10^6} \]
- Calculate:
\[ D = \sqrt{2 \times 20 \times 6.4 \times 10^6} = \sqrt{256 \times 10^6} = 16 \times 10^3 \text{ m} = 16 \text{ km} \]

3. **Calculate the area covered (A) for case (i):**
- The area covered is given by:
\[ A = \pi D^2 \]
- Substituting the value of D:
\[ A = \pi (16 \text{ km})^2 = \frac{22}{7} \times 256 \text{ km}^2 \approx 804.6 \text{ km}^2 \]

4. **Calculate the range (D1) for case (ii) where the receiving antenna is at a height of 25 m:**
- The formula for the range is modified to include both heights:
\[ D_1 = \sqrt{2 h_t R} + \sqrt{2 h_r R} \]
- Substituting the values:
\[ D_1 = \sqrt{2 \times 20 \times 6.4 \times 10^6} + \sqrt{2 \times 25 \times 6.4 \times 10^6} \]
- Calculate:
\[ D_1 = 16 \times 10^3 + \sqrt{2 \times 25 \times 6.4 \times 10^6} = 16 \times 10^3 + \sqrt{320 \times 10^6} = 16 \times 10^3 + 17.9 \times 10^3 = 33.9 \text{ km} \]

5. **Calculate the area covered (A1) for case (ii):**
- Using the same area formula:
\[ A_1 = \pi D_1^2 \]
- Substituting the value of D1:
\[ A_1 = \pi (33.9 \text{ km})^2 = \frac{22}{7} \times 1155.21 \text{ km}^2 \approx 3611.8 \text{ km}^2 \]

6. **Calculate the percentage increase in area covered:**
- The percentage increase is given by:
\[ \text{Percentage Increase} = \frac{A_1 - A}{A} \times 100 \]
- Substituting the values:
\[ \text{Percentage Increase} = \frac{3611.8 - 804.6}{804.6} \times 100 \approx 348.9\% \]

### Final Results:
- The area covered when the receiving antenna is at ground level is approximately **804.6 km²**.
- The area covered when the receiving antenna is at a height of 25 m is approximately **3611.8 km²**.
- The percentage increase in area covered when the antenna is at 25 m compared to ground level is approximately **348.9%**.