Chapter 9.3, Problem 7MRPS

Solution

To determine the area of each slice of pizza.

The area of first slice of pizza is $1080{\text{ in}}^2$ and the area of second slice of pizza is $1102.5{\text{ in}}^2.$

Given:

One slice of pizza has $6\text{ in}$ radius and ${60}^{\circ }$ arc angle. And the other slice of pizza has $7\text{ in}$ radius and ${45}^{\circ }$ arc angle.

Calculation:

The pizza slices have the shape of sectors. Now, calculating the area of the first pizza slice as:

  $\begin{array}{l}{A}_1=\frac{1}{2}{r}_1{}^2{\theta }_1\\ \Rightarrow A_1=\frac{1}{2}{\left(6\right)}^2\left(60\right)\\ \Rightarrow A_1=\frac{1}{2}\times 36\times 60\\ \Rightarrow A_1=\frac{2160}{2}\\ \Rightarrow A_1=1080{\text{ in}}^2\end{array}$

Next, calculating the area of the second pizza slice as:

  $\begin{array}{l}{A}_2=\frac{1}{2}{r}_2{}^2{\theta }_2\\ \Rightarrow A_2=\frac{1}{2}{\left(7\right)}^2\left(45\right)\\ \Rightarrow A_2=\frac{49\times 45}{2}\\ \Rightarrow A_2=\frac{2205}{2}\\ \Rightarrow A_2=1102.5{\text{ in}}^2\end{array}$

Hence, the first pizza slice area is $1080{\text{ in}}^2$ and the second slice area is $1102.5{\text{ in}}^2.$

To determine that which slice is the better deal.

The first slice will be the better deal.

Given:

One slice of pizza has $6\text{ in}$ radius and ${60}^{\circ }$ arc angle. And the other slice of pizza has $7\text{ in}$ radius and ${45}^{\circ }$ arc angle.

Calculation:

The cost of the first slice is $\$1.50$ and the cost of second slice is $\$\mathrm{1.70.}$

As the area of first slice is $1080{\text{ in}}^2$ , the cost of pizza per square inch is:

  $\begin{array}{l}{C}_1=\frac{1.50}{1080}\\ \Rightarrow C_1=0.00138\\ \Rightarrow C_1\approx 0.0014\end{array}$

The area of second slice is $1102.5{\text{ in}}^2$ , the cost of pizza per square inch is:

  $\begin{array}{l}{C}_2=\frac{1.70}{1102.5}\\ \Rightarrow C_2=0.00154\\ \Rightarrow C_2\approx 0.0015\end{array}$

The cost of pizza per square inch for first slice is less. Hence, the first slice will be the better deal.

To determine how could change the price of the $7\text{ inch}$ slice so that neither slice offers a better deal than the other.

The new price of the $7\text{ inch}$ slice is $\$1.54$

Given:

One slice of pizza has $6\text{ in}$ radius and ${60}^{\circ }$ arc angle. And the other slice of pizza has $7\text{ in}$ radius and ${45}^{\circ }$ arc angle.

Calculation:

The first pizza slice is a better deal and the cost per square inch of first pizza slice is $\$0.0014$ .

Next, the area for the second pizza slice is $\mathrm{1102.5.}$ To make the second pizza slice also a better deal, the new cost for the second pizza slice should be:

  $\begin{array}{l}{C}_S=\left(0.0014\right)\left(1102.5\right)\\ \Rightarrow C_S=1.5435\\ \Rightarrow C_S\approx \$1.54\end{array}$

Hence, the price of the $7\text{ inch}$ slice should be reduced to $\$1.54,$ then neither slice will offer a better deal than the other.