Question

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

• A. 12%
• B. 15%
• C. 14%
• D. 10%

Answer 1

The Correct Option is D

To determine the impact on the transmission range of a TV tower when its height is increased, we begin with the relationship between the height of the tower and its transmission range. The transmission range of a TV tower is directly proportional to the square root of its height. Mathematically, this is expressed as:

R \propto \sqrt{h}

Let the original height of the tower be h and the original range be R. The relationship can be given by:

R = k \sqrt{h}

where k is the proportionality constant.

When the height of the tower is increased by 21%, the new height h' is:

h' = h + 0.21h = 1.21h

The new transmission range R' is:

R' = k \sqrt{h'} = k \sqrt{1.21h}

Simplifying further:

R' = k \sqrt{1.21} \sqrt{h} = R \cdot \sqrt{1.21}

We need to calculate \sqrt{1.21}:

\sqrt{1.21} \approx 1.1

Therefore, the new range R' can be written as:

R' = 1.1R

The percentage increase in range is then given by:

\left(\frac{R' - R}{R} \times 100\%\right) = (1.1R - R) / R \times 100\% = 0.1 \times 100\% = 10\%

Thus, when the height of the TV tower is increased by 21%, its transmission range increases by 10%.