Answer
$h\approx11.55cm$, $r\approx8.16cm$. $V\approx2418.4cm^3$
Work Step by Step
Step 1. Draw a diagram as shown. The radius of the sphere is $R=10cm$, and the cylinder has a radius of $r$ and height $h$.
Step 2. Using the Pythagorean Theorem, we have $r^2+(\frac{h}{2})^2=R^2$ thus $r^2=10^2-(\frac{h}{2})^2$
Step 3. The volume of the cylinder is given by $V=\pi r^2 h=\pi h(100-\frac{h^2}{4})=100\pi h-\frac{\pi h^3}{4}$
Step 4. To find the maximum volume, take the derivative of the equation above; we have $V'=100\pi-\frac{3\pi h^2}{4}$
Step 5. Letting $V'=0$, we have $h^2=\frac{400}{3}$ and $h=\frac{20\sqrt {3}}{3}\approx11.55cm$ which gives $r^2=100-\frac{100}{3}=\frac{200}{3}$ and $r=\frac{10\sqrt 6}{3}\approx8.16cm$. The volume with these dimensions is $V=\pi r^2 h=\pi (\frac{200}{3}) (\frac{20\sqrt {3}}{3})=\frac{4000\pi\sqrt {3}}{9}\approx2418.4cm^3$
Step 6. Check $V''=-\frac{3\pi h}{2}\lt0$; thus, the function is concave down at this point and the volume is a maximum.
