Chapter 2.6, Problem 21AYU

21. Inscribing a Cylinder in a Cone Inscribe a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. see the illustration. Express the volume V of the cylinder as a function of r.

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

To determine

To find: To express the Volume $V$ of the cylinder as a function of $r$ .

Answer to Problem 21AYU

Solution:

$V\left(r\right)=\pi r^{2}H\left(\frac{R-r}{R}\right)$

Explanation of Solution

Given:

A right circular cylinder of height $h$ and radius $r$ is inscribed in a sphere of fixed radius $R$ and fixed height $H$.

Calculation:

To express the Volume $V$ of the cylinder as a function of $r$:

Volume of a cylinder $=\pi r^{2}h$.

From the figure, based on the properties of similar triangles, the ratio of sides can be written as

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

Therefore,

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

Hence, the volume of the cylinder is $=\pi r^{2}h$.

$=\pi r^2\left(H-\frac{r}{R}H\right)$

$= \pi r^{2}H\left(\frac{R-r}{R}\right)$