Chapter 12.7, Problem 12PPS

To determine

To find: the lateral and surface area

Answer to Problem 12PPS

The surface area of the cone is $\text{90}\text{.29}{\text{ft}}^2$ and the lateral area of the cone is $62.031{\text{ft}}^2$ .

Explanation of Solution

Given:

Consider the figure as shown below:

  

Formula used:

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

The surface area S of a cone is S = L + B

Calculation:

Consider the diameter of the cone is 6 ft and slant height of the cone is $6\frac{2}{3}$ ft

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

First convert the mixed number into decimal.

  $\begin{array}{c}6\frac{2}{3}=\frac{6\cdot 3+2}{3}\\ =\frac{18+2}{3}\\ =\frac{20}{3}\\ \approx 6.67\end{array}$

Find the lateral area of the cone.

Since radius is half the diameter of a circle, so radius of the circle is $\frac{6}{2}$ ft that is 3 ft.

Substitute r = 3 and l = 6.67 in $L=\pi rl$

Thus,

  $\begin{array}{c}L=\pi rl\\ =\pi \left(3\right)\left(6.67\right)\\ =20.01\pi \\ \approx 20.01\left(3.14\right)\left[Since\pi \approx 3.14\right]\\ =62.031\end{array}$

Therefore, the lateral area of the cone is approximately $62.031{\text{ft}}^2$ .

Find the surface area of the cone as follows:

The area B of the base is given by

  $\begin{array}{c}B=\pi r^2\\ =\pi {\left(3\right)}^2\\ =9\pi \\ \approx 9\left(3.14\right)\left[Since\pi \approx 3.14\right]\\ =28.26\end{array}$

Substitute $B\approx 28.26$ and $L\approx 62.031$ in S = L + B

  $\begin{array}{c}S=L+B\\ \approx 62.031+28.26\\ \approx \text{90}\text{.29}\end{array}$

Conclusion:

Therefore, the surface area of the cone is $\text{90}\text{.29}{\text{ft}}^2$ and the lateral area of the cone is $62.031{\text{ft}}^2$ .