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

Options

A. 0.1
B. 0.2
C. 0.3
D. 0.4

Answer

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

### Step 1: Understand the relationship between height and transmission range
The transmission range \( D \) of a TV tower is given by the formula:
\[ D = \sqrt{2hR} \]
where \( h \) is the height of the tower and \( R \) is a constant related to the environment.

### Step 2: Calculate the new height after a 21% increase
If the original height of the tower is \( h \), then after a 21% increase, the new height \( h' \) can be calculated as:
\[ h' = h + 0.21h = 1.21h \]

### Step 3: Find the new transmission range
Using the new height \( h' \), the new transmission range \( D' \) can be expressed as:
\[ D' = \sqrt{2h'R} = \sqrt{2(1.21h)R} = \sqrt{1.21} \cdot \sqrt{2hR} = \sqrt{1.21} \cdot D \]

### Step 4: Calculate the ratio of the new range to the original range
The ratio of the new range \( D' \) to the original range \( D \) is:
\[ \frac{D'}{D} = \sqrt{1.21} \]

### Step 5: Simplify the square root
Calculating \( \sqrt{1.21} \):
\[ \sqrt{1.21} = 1.1 \]
Thus, we have:
\[ \frac{D'}{D} = 1.1 \]

### Step 6: Determine the percentage increase in the transmission range
The percentage increase in the range can be calculated as:
\[ \text{Percentage Increase} = \left( \frac{D' - D}{D} \right) \times 100 \]
Substituting \( D' = 1.1D \):
\[ \text{Percentage Increase} = \left( \frac{1.1D - D}{D} \right) \times 100 = \left( \frac{0.1D}{D} \right) \times 100 = 10\% \]

### Conclusion
Therefore, the transmission range of the TV tower will increase by **10%** when the height of the tower is increased by 21%.

---