Question

If the radius of the circle is decreased by \[50\% \] then find the percentage decrease in area.

Answer 1

Hint: Clearly radius of a circle is directly proportional to the area of the circle because \[area = \pi {r^2}\]. Find a relation between the initial and final area that will help you to get the percent drop.
Complete Step by Step Solution:
We know that the area of the circle is given by \[\pi {r^2}\] where r is the radius
We are given that the radius is decreasing by \[50\% \] which means that it is becoming half
Therefore the new radius let it be R will be \[R = \dfrac{r}{2}\]
Therefore the new area becomes
\[\begin{array}{l}
\pi {R^2}\\
 = \pi {\left( {\dfrac{r}{2}} \right)^2}\\
 = \pi \left( {\dfrac{{{r^2}}}{4}} \right)\\
 = \dfrac{1}{4} \times \pi {r^2}
\end{array}\]
Now if we let that the initial area of the circle was denoted by \[a\] and the final area is denoted by \[A\]
Then the relation thus establishes is
\[\begin{array}{l}
\therefore A = \dfrac{1}{4} \times a\\
 \Rightarrow \dfrac{A}{a} = \dfrac{1}{4}
\end{array}\]
 Clearly the final area is \[\dfrac{1}{4}\] times of the initial area which means if the initial area was 100% then the final area is 25%
From here we can say that its a drop of 75%
Therefore the area will decrease by 75%

Note: After getting the relation we can also subtract it from the initial i.e., as \[A = 0.25 \times a\] then if we subtract from initial it will be \[a - 0.25 \times a = 0.75 \times a\] so from here also it can be easily seen that the area is decreased by 75%