Chapter 0.2, Problem 54E

a.

To determine

To find : The volume of a cylinder as a function of height, where height $h$ is equal to diameter .

Answer to Problem 54E

The volume of the function in term of height $V\left(h\right)=\pi \frac{h^3}{4}.$

Explanation of Solution

Given information:

The height $h$ is equal to diameter .

Concept used:

The volume of a cylinder $V=\pi r^{2}h.$ .

Calculation:

Here height $h$ is equal to diameter $h=D$ .

Then radius is $r=\frac{h}{2}\sin cer=\frac{D}{2}$ .

Substitute $r$ and height into the equation of volume ,

  $\begin{array}{c}V=\pi r^{2}h.\\ =\pi \frac{h^2}{4}\times h\\ =\frac{\pi h^3}{4}\end{array}$

Thus the volume of the function in term of height $V\left(h\right)=\pi \frac{h^3}{4}.$

b.

To determine

To find : The surface of a cylinder as a function of height, where height $h$ is equal to diameter .

Answer to Problem 54E

The volume of the function in term of height $V\left(h\right)=\pi \frac{h^3}{4}.$

Explanation of Solution

Given information:

The height $h$ is equal to diameter .

Concept used:

The surface area of a cylinder $S=2\pi r^2+2\pi rh.$ .

Calculation:

Here height $h$ is equal to diameter $h=D$ .

Then radius is $r=\frac{h}{2}\sin cer=\frac{D}{2}$ .

Substitute $r$ and height into the equation of surface area ,

  $\begin{array}{c}S=2\pi r^2+2\pi rh.\\ =2\pi \frac{h^2}{4}+\pi h^2\\ =\frac{3\pi h^2}{2}\end{array}$

Thus the surface of the function in term of height $S\left(h\right)=\frac{3\pi h^2}{2}.$

c.

To determine

To find : The volume of a cylinder where surface area is $54\pi$ square inch.

Answer to Problem 54E

The volume is $54\pi$ cubic inches.

Explanation of Solution

Given information:

The surface area of cylinder $54\pi$ square inch.

Concept used:

The surface area of a cylinder $S\left(h\right)=\frac{3\pi h^2}{2}.$

The volume of a cylinder $V\left(h\right)=\pi \frac{h^3}{4}.$

Calculation:

The surface area of a cylinder is $54\pi$ square inch.

That is

  $\begin{array}{c}S\left(h\right)=\frac{3\pi h^2}{2}\\ \frac{3\pi h^2}{2}=54\pi \\ h=6\text{inches}\text{.}\end{array}$

Substitute $h=6\text{inches }$ equation of volume,

  $\begin{array}{c}V\left(h\right)=\pi \frac{h^3}{4}.\\ =\frac{\pi 6^3}{4}\\ =54\pi \text{cubic}\text{inches}\text{.}\end{array}$

Thus the volume is $54\pi$ cubic inches.