Chapter 1, Problem 67RE

Solution

(a).

To write: The height $h$ as a function of radius $r$ .

The height $h$ as a function of radius $r$ is equal to $2\sqrt{3-r^2}$ .

Given information:

A right circular cylinder of radius $r$ and the height $h$ is inscribed inside a sphere of radius $\sqrt{3}$ inches.

  

Calculation:

Use the Pythagorean Theorem in the right triangle and calculate the height below:

  $\begin{array}{c}{\left(\sqrt{3}\right)}^2={\left(\frac{h}{2}\right)}^2+{\left(r\right)}^2\\ 3=\frac{h^2}{4}+r^2\\ \frac{h^2}{4}=3-r^2\\ h^2=4\left(3-r^2\right)\\ h=\sqrt{4\left(3-r^2\right)}\\ h=2\sqrt{3-r^2}\end{array}$

Therefore, the height $h$ as a function of radius $r$ is $2\sqrt{3-r^2}$ .

(b).

To write: The volume $V$ of the cylinder as a function of radius $r$ .

The volume $V$ of the cylinder is equal to $2\pi r^2\sqrt{3-r^2}$ .

Given information:

A right circular cylinder of radius $r$ and the height $h$ is inscribed inside a sphere of radius $\sqrt{3}$ inches.

  

Calculation:

From the part (a),

  $h=2\sqrt{3-r^2}$

Formula used:

The volume of the cylinder is given by

  $V=\pi r^{2}h$

Substitute $h=2\sqrt{3-r^2}$ in the above formula and calculate the volume of the cylinder in terms of $r$ below:

  $\begin{array}{c}V=\pi r^2\left(2\sqrt{3-r^2}\right)\\ =2\pi r^2\sqrt{3-r^2}\end{array}$

Therefore, the volume $V$ of the cylinder is $2\pi r^2\sqrt{3-r^2}$ .

(c).

To find: The values of $r$ in the domain of $V$ .

The values of $r$ are belong to $\left[-\sqrt{3},\sqrt{3}\right]$ .

Given information:

A right circular cylinder of radius $r$ and the height $h$ is inscribed inside a sphere of radius $\sqrt{3}$ inches.

  

Calculation:

From the part (b),

  $V=2\pi r^2\sqrt{3-r^2}$

For the domain of the volume, substitute the square root function is greater than or equal to zero and solve for $r$ :

  $\begin{array}{c}3-r^2\ge 0\\ r^2\le 3\\ r\le \pm \sqrt{3}\\ r\in \left[-\sqrt{3},\sqrt{3}\right]\end{array}$

Therefore, the values of $r$ are belong to $\left[-\sqrt{3},\sqrt{3}\right]$ .

(d).

To sketch: The graph $V$ over the domain $\left[0,\sqrt{3}\right]$ .

The graph of the volume $V$ is

  

Given information:

A right circular cylinder of radius $r$ and the height $h$ is inscribed inside a sphere of radius $\sqrt{3}$ inches.

  

Calculation:

From the part (b),

  $V=2\pi r^2\sqrt{3-r^2}$

Now, draw the graph of the volume $V$ over the domain $\left[0,\sqrt{3}\right]$

below

(e).

To find: The maximum volume from the graph.

The maximum volume is equal to $3\pi$ inches³.

Given information:

A right circular cylinder of radius $r$ and the height $h$ is inscribed inside a sphere of radius $\sqrt{3}$ inches.

  

Calculation:

From the part (d),

The graph of the volume is below:

  

From the above graph, the maximum volume is $3\pi$ inches³.

Therefore, the maximum volume is $3\pi$ inches³.