Answer
Interval: $[\sqrt[3]{\frac{6}{\pi}}, \sqrt[3]{\frac{9}{\pi}}]$
Work Step by Step
We know that the volume $V$ is restricted to fall between $8$ and $12$:
$8 \leq\frac{4}{3}\pi r^{3}\leq 12$
We solve this inequality for $r$:
$8*\frac{3}{4\pi} \leq r^{3}\leq 12*\frac{3}{4\pi}$
$\frac{6}{\pi}\leq r^{3}\leq\frac{9}{\pi}$
$\sqrt[3]{\frac{6}{\pi}}\leq r\leq\sqrt[3]{\frac{9}{\pi}}$
Interval: $[\sqrt[3]{\frac{6}{\pi}}, \sqrt[3]{\frac{9}{\pi}}]$
