Chapter 12.7, Problem 31SR

Solution

To find: the lateral and total surface area.

The lateral area is $386.22i{n}^2$ .

Total surface area is $\text{423}{\text{.9 in}}^2.$

Given: radius = 6 inches, height ¼ inches.

Formulas used:

The lateral area $\text{L}$ of a cylinder is $\text{L=2}\pi rh$

The surface area $S$ of rectangular prism is $S=L+2B$

Calculation:

Consider the radius of the cylinder is $6in$ , and the height of the cylinder is $10\frac{1}{4}in.$

The objective is to find the lateral and surface area of the cylinder.

Convert the mixed number into decimal.

  $\begin{array}{l}h=10\frac{1}{4}\\ \text{ =}\frac{10.4+1}{4}\\ \text{ =}\frac{40+1}{4}\\ \text{ =}\frac{41}{4}\\ \text{ =10}\text{.25}\end{array}$

Find the lateral area of the cylinder as follows:

Substitute $r=6$ and $h=10.25$ in $\text{L=2}\pi rh$

Thus,

  $\begin{array}{l}\text{L=2}\pi rh\\ \text{L=2}\pi (6)(10.25)\\ \text{ =2}\pi (61.5)\\ \text{ =123}\pi \\ \approx 123(3.14)\\ \text{ =386}\text{.22}\end{array}$ As $\pi =3.14$

Hence, the lateral area of the cylinder is approximately $386.22i{n}^2$ .

Find the surface area of the cylinder as follows:

The area $B$ of the base is $B=\pi r^2$

Here, $r$ is the radius of the circle.

Substitute $r=6$ in $B=\pi r^2$

  $\begin{array}{l}B=\pi r^2\\ $ B=\pi {(6)}^2$\text{ =36}\pi \\ \approx 6(3.14)\\ \approx \text{18}\text{.84}\end{array}$ As $\pi =3.14$

Substitute $B\approx 18.84$ and $L\approx 386.22$ in $S=L+2B$

  $\begin{array}{l}S=L+2B\\ \approx \text{386}\text{.22+2(18}\text{.84)}\\ \text{ =386}\text{.22+37}\text{.68}\\ \text{ =423}\text{.9}\end{array}$

Hence, the surface area of the cylinder is approximately $\text{423}{\text{.9 in}}^2.$