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: $3.5\;\mathrm{cm}$
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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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