The sum of the areas of two circles is $290 \pi$ square inches. The length of a radius of the smaller circle is 2 inches less than the length of a radius of the larger circle. Find the length of a radius of the larger circle.
A) 15 in.
B) 11 in.
C) 13 in.
D) 12 in.

Question

Quizplus · Accepted Answer

Let the radius of the larger circle be $r$, so the radius of the smaller circle is $r-2$. The sum of the areas of the two circles is $\pi r^2+\pi (r-2)^2 = 2\pi (r^2-2r+4)$. We set this equal to $290\pi$ and solve: \begin{align*}
2\pi(r^2-2r+4)&=290\pi\
\Rightarrow\qquad r^2-2r+4&=145\
\Rightarrow\qquad r^2-2r-141&=0\
\Rightarrow\qquad (r-13)(r+11)&=0
\end{align*}Since $r$ is the length of the radius of the larger circle, we take $r=13$. Therefore, the answer is $\boxed{	extbf{(C)}}$.

The Sum of the Areas of Two Circles Is 290π290 \pi290π

The Sum of the Areas of Two Circles Is $290 \pi$