For Exercises 2–3, refer to the following: The area of a circle of radius is . Expressed in terms of the diameter, , the area is . If we apply a linear scaling factor to the diameter, then the area of the scaled circle—just as is the case for a square—changes with , the square of the linear scaling factor.

Question 18.32

2. A natural application of this idea is to pizza. The prices at Vince’s pizza restaurant in Beloit, Wisconsin, are $7.57, $8.71, $9.49, $11.13, and $12.63, respectively, for small (10-in.), medium (12-in.), large (14-in.), extra-large (16-in.), and XX-large (18-in.) cheese pizzas.

  1. What is the linear scaling factor for an XX-large pizza compared with a small one?
  2. How many times as large in area is the extra-large pizza compared with the small one?
  3. How much pizza does each size give per dollar? What “hidden” assumptions are you making about how the pizzas are scaled up?
  4. The corresponding prices for a pizza with six toppings are $12.35, $15.49, $18.42, $21.60, and $25.74. Is there any size of these for which you get more pizza per dollar than some size of the cheese pizzas? (Curiously, all the prices are, to the nearest cent, exactly $0.10 higher than four years earlier! That is an example of arithmetic scaling.)