Chapter 11.5, Problem 15PPS

(a)

To determine

To find: The side length of the smaller pan.

Answer to Problem 15PPS

The side length of the smaller pan is $4$ inches.

Explanation of Solution

Given:

The side length of the larger pan is $4.5\text{in}$ and the area of the smaller pan is $41.6{\text{in}}^2$ .

Calculation:

Consider the formula to determine the area of the regular hexagon is,

  $S=\frac{n}{4\tan \left(\frac{180°}{n}\right)}{s}^2$

Consider the given diagram as shown in Figure 1

  

Figure 1

Then, by the tangent ratio to find the apothem.

  $\begin{array}{l}\tan \left(\frac{180°}{n}\right)=\frac{\frac{s}{2}}{a}\\ a=\frac{\frac{s}{2}}{\tan \left(\frac{180°}{n}\right)}\end{array}$

Then, the area is,

  $\begin{array}{c}A=\frac{1}{2}aP\\ =\frac{1}{2}\left(\frac{\frac{s}{2}}{\tan \left(\frac{180°}{n}\right)}\right)\left(ns\right)\\ =\frac{n}{4\tan \left(\frac{180°}{n}\right)}{s}^2\end{array}$

Then, determine the side length by substituting the area as,

  $\begin{array}{l}41.6=\frac{6}{4\tan \left(\frac{180°}{6}\right)}{s}^2\\ s^2=\frac{2\left(41.6\right)\tan 30°}{3}\\ s=16.01\\ s=4\end{array}$

Thus, the side length of the smaller pan is $4$ inches.

(b)

To determine

To find: The pan that should be chosen.

Answer to Problem 15PPS

The area of the larger pan is $52.6{\text{in}}^2$ that is close to the area of the circular pan so the larger pan must be used.

Explanation of Solution

Given:

The diameter of the circular pan is 8 inches.

Calculation:

Consider the area of the pan is calculated as,

  $\begin{array}{c}A=\pi {\left(\frac{d}{2}\right)}^2\\ =\pi {\left(\frac{8}{2}\right)}^2\\ =16\pi \\ =50.3{\text{in}}^2\end{array}$

Since, the side of the larger pan is 4.5 inches. Then,

  $\begin{array}{c}S=\frac{6}{4\tan \left(\frac{180°}{6}\right)}{\left(4.5\right)}^2\\ =\frac{3}{2\tan 30°}{\left(4.5\right)}^2\\ =52.6{\text{in}}^2\end{array}$

Thus, the area of the larger pan is $52.6{\text{in}}^2$ that is close to the area of the circular pan so the larger pan must be used.