Chapter 12.7, Problem 16PPS

To determine

To find: the surface area.

Answer to Problem 16PPS

Total surface area is $\text{137}{\text{.5 in}}^2.$

Explanation of Solution

Given: Consider the figure as shown below:

  

Formulas used:

The lateral area $\text{L}$ of the cone is $\text{L=}\pi \text{rl}$

Calculation:

The figure is combination of two cones.

The perimeter of each cone is $2\times 5in$ .that is $10\text{ in}$ and the slant height of the first cone is $\text{12}\text{.3 in}$ and the slant height of the second cone is $\text{15}\text{.2 in}$

The objective is to find the surface area of the figure.

The surface area of the figure is the sum of the lateral area of cone.

Find the lateral area of the first cone as follows:

To find the lateral area find the radius of the circle.

The perimeter $\text{P}$ of a circle is $\text{P =2}\pi \text{r}$

  $\begin{array}{l}\text{P =2}\pi \text{r}\\ \text{10=2}\pi \text{r}\\ \frac{10}{2}=\pi \text{r}\\ 5=\pi \text{r}\\ \frac{5}{\pi }=r\\ r=\frac{5}{\pi }\end{array}$

Substitute $r=\frac{5}{\pi }$ and $l_1=12.3$ in $L_1=\pi r{l}_1$

Thus

  $\begin{array}{l}{L}_1=\pi r{l}_1\\ \text{ =}\pi .\frac{5}{\pi }(12.3)\\ \text{ =5(12}\text{.3)}\\ \text{ =61}\text{.5}\end{array}$

Therefore, the lateral area of first cone is $\text{61}{\text{.5 in}}^2$

Find the lateral area of the second cone as follows:

Put $r=\frac{5}{\pi }$ and $l_2=15.2$ in the lateral area formula to find the lateral area $L_2$ of the second cone.

Thus,

  $\begin{array}{l}{L}_2=\pi r{l}_2\\ \text{ =}\pi .\frac{5}{\pi }(15.2)\\ \text{ =5(15}\text{.2)}\\ \text{ =76}\end{array}$

Therefore, the lateral area of the second cone is ${\text{76 in}}^2$

The surface area $S$ is

  $\begin{array}{l}S=L_1+L_2\\ \text{ =61}\text{.5+76}\\ \text{ =137}\text{.5}\end{array}$

Conclusion:

Hence the surface area of the figure is approximately $\text{137}{\text{.5 in}}^2.$