Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

Inscribe a right circular cylinder of height h and radius r in a sphere of fixed radius R. See the illustration. Express the volume V of the cylinder as a function of h.

[Hint: $V={\mathrm{\pi r}}^2\mathrm{h}$. Note also the right triangle.]

Step-by-Step Solution

Answer

The volume of the cylinder as a function of h is $V=\frac{\mathrm{\pi }(4{\mathrm{R}}^2-{\mathrm{h}}^2)\mathrm{h}}{4}$.

Step 1. Using Pythagorean Theorem in the right triangle.

$$\begin{gathered} r^2+{\left(\frac{h}{2}\right)}^2=R^2 \\ {r}^2=R^2-\frac{h^2}{4} \\ {r}^2=\frac{4R^2-h^2}{4} \end{gathered}$$

Step 2. Use the formula for the volume of a cylinder.

Volume of a cylinder is, $V={\mathrm{\pi r}}^2\mathrm{h}$.

Substitute the values.

$V=\mathrm{\pi }\left(\frac{4{\mathrm{R}}^2-{\mathrm{h}}^2}{4}\right)\mathrm{h}$

The volume V of the cylinder as a function of h is $V=\frac{\mathrm{\pi }(4{\mathrm{R}}^2-{\mathrm{h}}^2)\mathrm{h}}{4}$.