Question

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

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

Short answer

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

Step-by-step solution

Step 1. Use the property of similar triangles.

We can see two similar triangles in the figure.

Since the two triangles are similar, the corresponding sides are proportional.

$\frac{r}{R}=\frac{H-h}{H}$

Step 2. Simplify the equation to find the value of h.

Isolate h to one side.

$$\begin{gathered} rH=R(H-h) \\ rH=RH-Rh \\ Rh=RH-rH \\ Rh=(R-r)H \\ h=\frac{(R-r)H}{R} \end{gathered}$$

Step 3. Find the volume of the cylinder using the formula for the volume.

The volume V as a function of r is

$$\begin{gathered} V={\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{V}={\mathrm{\pi r}}^{\mathrm{r}}\frac{(\mathrm{R}-\mathrm{r})\mathrm{H}}{\mathrm{R}} \end{gathered}$$