Solutions

Let $A_s(t)$ and $A_b(t)$ be the areas of the small and big circles respectively, given time $t$. These functions are given by $A_s(t)=\pi r_s(t)^2$ and $A_b(t)=\pi r_b(t)^2$, where $r_s(t)$ and $r_b(t)$ are the radii of the small and big circles respectively, given time $t$. Let $C_s(t)=2\pi r_s(t)$ denote the circumference of the small circle.

We have $A_b(t)-A_s(t)=9$. Hence, $A_s(t)=A_b(t)-9$ and $A_s'(t)=A_b'(t)$. This makes intuitive sense since the area between the two circles is constant. Moreover,

$$r_s(t)=\sqrt{\frac{A_s(t)}{\pi}},$$

so

$$r_s'(t)=\frac{1}{2\sqrt{A_s(t)\pi}}A_s'(t).$$

Hence,

$$\begin{align*} C_s'(t)&=2\pi r_s'(t) \\ &=2\pi\left(\frac{1}{2\sqrt{A_s(t)\pi}}A_s'(t)\right) \\ &=2\pi\left(\frac{1}{2\sqrt{16\pi^2}}\cdot 10\pi\right) \\ &=\frac{10\pi}{4} \\ &=\frac{5\pi}{2}. \end{align*}$$