Question

If the radius of a circle is increased by 20% then the corresponding increase in the area of the circle is _______?
\[
  {\text{A}}{\text{. 40% }} \\
  {\text{B}}{\text{. 44% }} \\
  {\text{C}}{\text{. 20% }} \\
  {\text{D}}{\text{. 21% }} \\
\]

Answer 1

Hint:In order to find the corresponding increase in the area of the circle, we start off by taking the radius of the circle as some random variable, calculate the new radius and substitute it in the formula of area of circle to find the increase in area.

Complete step-by-step answer:
Given Data,
Radius of circle is increased by 20%
Let us consider the initial radius of the circle to be some variable ‘x’.
We know the area of a circle of radius r is given by the formula,
 $ {\text{A = 4}}\pi {{\text{r}}^2} $
Here in this case, our radius is r = x.
Therefore the initial area of the circle is:
 $
   \Rightarrow {\text{A = 4}}\pi {\left( {\text{x}} \right)^2} \\
   \Rightarrow {\text{A = 4}}\pi {{\text{x}}^2} \\
 $

Given that the radius is increased by 20%,
⟹New radius R = x + 20% of x
 $
   \Rightarrow {\text{R = x + }}\dfrac{{20}}{{100}} \times {\text{x}} \\
   \Rightarrow {\text{R = }}\dfrac{{100{\text{x + }}20{\text{x}}}}{{100}} \\
   \Rightarrow {\text{R = }}\dfrac{{120{\text{x}}}}{{100}} = 1.2{\text{x}} \\
 $
Therefore the new area of the circle becomes:
 $ {\text{A = 4}}\pi {{\text{R}}^2} $ , where R = 1.2x
 $
   \Rightarrow {\text{A = 4}}\pi {\left( {1.2{\text{x}}} \right)^2} \\
   \Rightarrow {\text{A = 4}}\pi \left( {1.44{{\text{x}}^2}} \right) \\
 $

The increase of area of the circle = New area – initial area
⟹Increase in area = $ {\text{4}}\pi \left( {1.44{{\text{x}}^2}} \right) - {\text{4}}\pi \left( {{{\text{x}}^2}} \right) $
⟹Area Increase = $ {\text{4}}\pi \left( {1.44 - 1} \right){{\text{x}}^2} = 4\pi {{\text{x}}^2}\left( {0.44} \right) $
We know the formula of percentage increase is given by,
 $ {\text{Percent Increase = }}\dfrac{{{\text{difference}}}}{{{\text{Original}}}} \times 100 $
Therefore the percentage increase in the area of circle is given by,
 $
   \Rightarrow {\text{Percent Increase of area = }}\dfrac{{4\pi {{\text{x}}^2}\left( {0.44} \right)}}{{4\pi {{\text{x}}^2}}} \times 100 \\
   \Rightarrow {\text{Percent Increase of area = }}\left( {0.44} \right) \times 100 = 44\% \\
 $

Therefore the corresponding increase in the area of the circle is 44%.
Option B is the correct answer.

Note:In order to solve this type of problems the key is to know the formulae of the area of the circle. Knowing the concept of percentages and percent change is very important. Given the radius is increased by 20% for which we added 20% of the initial to the original value to find the new value.We calculated percent change of the area because all the answer choices are given in percentages. The sign of the difference in calculating percent change does not matter, it is always positive. Difference is generally calculated by subtracting the original value from the new value.