Question

A TV tower has a height of 100m. By how much the height of the tower should be increased to triple its coverage range (in m).
A. 200
B. 800
C. 900
D. 300

Answer 1

Hint: Coverage range of a TV tower is the geographic area where the tower can communicate. The question talks about the height of a certain television tower which is installed and erected to stand and how the height would be if allowed to be clamped on each other in three different pieces. Coverage range is directly proportional to the square root of the height of the tower. Use this info to solve the question.

Complete step by step answer:
We are given that the height of a TV tower is 100m.
We have to find the height of the tower when the coverage range is tripled.
Let the coverage range be R.
Let the new coverage area be ${R_o}$
By the given info, we can say that the new coverage area is three times (triple) the older coverage area
${R_o} = 3 \times R = 3R$
Coverage range is directly proportional to the square root of height.
$
  R \propto \sqrt H \\
   \implies \dfrac{R}{{{R_o}}} = \dfrac{{\sqrt H }}{{\sqrt {{H_o}} }} \\
 $
Where H is equal to 100m
$
  \dfrac{R}{{{R_o}}} = \dfrac{{\sqrt {100} }}{{\sqrt {{H_o}} }} \\
  {R_o} = 3R \\
   \implies \dfrac{R}{{3R}} = \sqrt {\dfrac{{100}}{{{H_o}}}} \\
   \implies {\left( {\dfrac{R}{{3R}}} \right)^2} = \dfrac{{100}}{{{H_o}}} \\
   \implies \dfrac{{{R^2}}}{{9{R^2}}} = \dfrac{{100}}{{{H_o}}} \\
   \implies {H_o} = 100 \times 9 = 900m \\
 $
Therefore, when the height of the tower is 900 m then the coverage range gets tripled.
The increase in height should be $900 - 100 = 800m$
Therefore, the height of the tower should be increased by 800 m to triple its coverage area.

So, the correct answer is “Option B”.

Note:
The height of a transmitting antenna or a tower is directly proportional to the square of the distance covered (coverage). Greater the height of an antenna, greater is its coverage. Example for coverage range is the area within which a phone will finish calls using a mobile network.