By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by `21%`?

Options

A. `15%`
B. `12%`
C. `10%`
D. `14%`

Answer

To solve the problem of how the transmission range of a TV tower is affected by an increase in its height by 21%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for transmission range**:
The transmission range \( D \) of a TV tower is given by the formula:
\[ D = \sqrt{2Rh} \]
where \( R \) is a constant and \( h \) is the height of the tower.

2. **Calculate the new height**:
If the height of the tower is increased by 21%, the new height \( h' \) can be expressed as:
\[ h' = h + 0.21h = 1.21h \]

3. **Calculate the new transmission range**:
The new transmission range \( D' \) with the increased height will be:
\[ D' = \sqrt{2R(1.21h)} = \sqrt{2R} \cdot \sqrt{1.21h} \]
This can be simplified to:
\[ D' = \sqrt{2Rh} \cdot \sqrt{1.21} = D \cdot \sqrt{1.21} \]

4. **Find the percentage change in transmission range**:
The percentage change in transmission range can be calculated using the formula:
\[ \text{Percentage Change} = \frac{D' - D}{D} \times 100 \]
Substituting \( D' \) into the equation:
\[ \text{Percentage Change} = \frac{D \cdot \sqrt{1.21} - D}{D} \times 100 \]
Simplifying this gives:
\[ \text{Percentage Change} = (\sqrt{1.21} - 1) \times 100 \]

5. **Calculate \( \sqrt{1.21} \)**:
We know that:
\[ \sqrt{1.21} = 1.1 \]
Therefore:
\[ \text{Percentage Change} = (1.1 - 1) \times 100 = 0.1 \times 100 = 10\% \]

### Final Answer:
The transmission range of the TV tower will be affected by an increase of **10%** when the height of the tower is increased by 21%.