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 an increase in the height of the tower by 21%, we can follow these steps:

### Step 1: Understand the relationship between height and range
The transmission range \( D \) of a TV tower is given by the formula:
\[ D = 2 \sqrt{h_t \cdot R_e} \]
where:
- \( h_t \) = height of the transmitting tower
- \( R_e \) = radius of the Earth

### Step 2: Define the initial and new heights
Let the initial height of the tower be \( h_{t1} \). If the height is increased by 21%, the new height \( h_{t2} \) can be expressed as:
\[ h_{t2} = h_{t1} + 0.21 h_{t1} = 1.21 h_{t1} \]

### Step 3: Calculate the initial and new ranges
Using the formula for range, we can express the initial range \( D_1 \) and the new range \( D_2 \):
\[ D_1 = 2 \sqrt{h_{t1} \cdot R_e} \]
\[ D_2 = 2 \sqrt{h_{t2} \cdot R_e} = 2 \sqrt{(1.21 h_{t1}) \cdot R_e} \]

### Step 4: Simplify the new range
Substituting \( h_{t2} \) into the equation for \( D_2 \):
\[ D_2 = 2 \sqrt{1.21 h_{t1} \cdot R_e} = 2 \sqrt{1.21} \sqrt{h_{t1} \cdot R_e} \]
This can be simplified to:
\[ D_2 = \sqrt{1.21} \cdot D_1 \]

### Step 5: Calculate the percentage increase in range
To find the percentage increase in range, we need to determine the ratio of the new range to the old range:
\[ \frac{D_2}{D_1} = \sqrt{1.21} \]
Calculating \( \sqrt{1.21} \):
\[ \sqrt{1.21} \approx 1.1 \]
This means:
\[ D_2 \approx 1.1 D_1 \]
The percentage increase in range can be calculated as:
\[ \text{Percentage Increase} = \left( \frac{D_2 - D_1}{D_1} \right) \times 100 = \left( \frac{1.1 D_1 - D_1}{D_1} \right) \times 100 = 0.1 \times 100 = 10\% \]

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

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