Question : The area of a circle is increased by $22\;\mathrm{cm}$, and its radius is increased by $1\;\mathrm{cm}$. The original radius of the circle is:
Option 1: $6\;\mathrm{cm}$
Option 2: $3.2\;\mathrm{cm}$
Option 3: $3\;\mathrm{cm}$
Option 4: $3.5\;\mathrm{cm}$
Correct Answer: $3\;\mathrm{cm}$
Solution :
The area of a circle with $r$ as the original radius $= \pi r^2$
Given that the area of the circle is increased by $22\;\mathrm{cm^2}$ and the original radius is increased by $1\;\mathrm{cm}$.
The area of the circle after the radius is increased $=\pi (r+1)^2$
$⇒\pi (r+1)^2 = \pi r^2 + 22$
$⇒\pi r^2 + 2\pi r + \pi = \pi r^2 + 22$
$⇒2\pi r + \pi = 22$
$⇒(2 r + 1)\pi = 22$
$⇒(2 r + 1) = \frac{22}{\pi}$
$⇒2 r + 1 = 7$
$⇒r = 3$
Hence, the correct answer is $3\;\mathrm{cm}$.
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