By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by 21% ?
- 12%
- 15%
- 14%
- 10%
The Correct Option is D
Solution and Explanation
TV Tower Transmission Range Problem
Step 1: Transmission Range Formula
The transmission range (\( d \)) of a TV tower is given by:
\( d = \sqrt{2Rh} \)
where \( R \) is the radius of the Earth and \( h \) is the height of the tower.
Step 2: New Height
If the height is increased by 21%, the new height (\( h' \)) is:
\( h' = h + 0.21h = 1.21h \)
Step 3: New Transmission Range
The new transmission range (\( d' \)) is:
\( d' = \sqrt{2Rh'} = \sqrt{2R(1.21h)} = \sqrt{1.21} \sqrt{2Rh} = 1.1\sqrt{2Rh} \)
Step 4: Percentage Increase
Since \( d = \sqrt{2Rh} \), the new range is:
\( d' = 1.1d \)
The percentage increase in the transmission range is:
\( \frac{d' - d}{d} \times 100 = \frac{1.1d - d}{d} \times 100 = 0.1 \times 100 = 10\% \)
Conclusion:
The transmission range increases by 10% (Option 4).
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