### Instruction: A $12^{\prime \prime}$-diameter pizza and a $16^{\prime \prime}$-diameter pizza are each cut into eight congruent slices. Jane ate three slices of the $12^{\prime \prime}$ pizza. Mark ate three slices of the $16^{\prime \prime}$ pizza. How many more square inches of pizza did Mark eat than Jane? Express your answer as a common fraction in terms of $\pi$.### Response: To compare the areas of the slices, I need to find the area of each whole pizza and then divide by eight.
The area of a circle is $\pi r^2$, where $r$ is the radius.
The radius of the $12^{\prime \prime}$ pizza is half of the diameter, so $r = 6^{\prime \prime}$.
The area of the $12^{\prime \prime}$ pizza is $\pi (6^{\prime \prime})^2 = 36\pi \text{ in}^2$.
The area of one slice of the $12^{\prime \prime}$ pizza is $\frac{36\pi \text{ in}^2}{8} = 4.5\pi \text{ in}^2$.
Jane ate three slices, so her total area is $3 \times 4.5\pi \text{ in}^2 = 13.5\pi \text{ in}^2$.
Similarly, the radius of the $16^{\prime \prime}$ pizza is $8^{\prime \prime}$, and the area of the whole pizza is $\pi (8^{\prime \prime})^2 = 64\pi \text{ in}^2$.
The area of one slice of the $16^{\prime \prime}$ pizza is $\frac{64\pi \text{ in}^2}{8} = 8\pi \text{ in}^2$.
Mark ate three slices, so his total area is $3 \times 8\pi \text{ in}^2 = 24\pi \text{ in}^2$.
To find the difference in areas, I subtract Jane's area from Mark's area.
The difference is $24\pi \text{ in}^2 - 13.5\pi \text{ in}^2 = 10.5\pi \text{ in}^2$.