Chapter 12.3, Problem 17PPS

To determine

To find:

The lateral area and surface area of the cone.

Answer to Problem 17PPS

The lateral area of cone is $\text{241}{\text{.62 ft}}^2$ and the surface area is $\text{447}{\text{.6 ft}}^2$ .

Explanation of Solution

Given information:

A cone with altitude is $\text{5 ft}$ and the slant height is $\text{9}\frac{1}{2}\text{ ft}$ .

Calculation:

The lateral area of cone is $L=\pi rl$ , and the surface area of cone is $S=L+\pi r^2$ where $r$ is the radius of the base and $l$ be the slant height.

Find the radius of the base of a cone.

Apply this formula to find the radius.

  $\begin{array}{l}{l}^2=r^2+h^2\\ r^2=l^2-h^2\\ r=\sqrt{l^2-h^2}\end{array}$

Substitute $h=5$ and $l=\frac{19}{2}$ in the above formula.

  $\begin{array}{c}r=\sqrt{{\left(\frac{19}{2}\right)}^2-5^2}\\ =\sqrt{\frac{361}{4}-25}\\ =\sqrt{\frac{361-100}{4}}\\ =\sqrt{\frac{261}{4}}\\ =\frac{16.15}{2}\\ =8.075\\ \approx 8.1\end{array}$

So, the radius of the base of a cone is $\text{8}\text{.1 cm}$ .

The lateral area of cone is $L=\pi rl$ .

  $\begin{array}{c}L=\left(\pi \right)\left(8.1\right)\left(\frac{19}{2}\right)\\ =\left(\pi \right)\left(4.05\right)\left(19\right)\\ =\left(3.14\right)76.95\\ =241.62\end{array}$

The surface area of cone is $S=L+\pi r^2$ .

  $\begin{array}{c}S=241.62+\pi {\left(8.1\right)}^2\\ =241.62+65.61\pi \\ =241.62+65.61\left(3.14\right)\\ =241.62+206.0154\\ =447.6354\\ \approx 447.6\end{array}$

Hence, the lateral area of cone is $\text{241}{\text{.62 ft}}^2$ and the surface area is $\text{447}{\text{.6 ft}}^2$ .