How to find how fast the circumference of a concentric circle is changing?

**citcat** · September 1st 2012, 12:02 PM

In this problem there are two varying concentrical circles with a constant area between at 9pi in^2. The area of the larger circle has a rate of change of 10pi in^2/sec. How fast is the circumference of the smaller circle changing when it has an area of 16pi in^2?
Thanks!

**HallsofIvy** · September 1st 2012, 04:53 PM

Let R be the area of the larger circle and r the area of the smaller circle. Then the area between them is $\pi R^2- \pi r^2= \pi(R^2- r^2)= 9\pi$ . Now differentiate both sides with respect to t and use the fact that $\frac{dR}{dt}= 10\pi$ and $\pi r^2= 16\pi$ . What is R at that time?

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