The Problem
When Thomas enters a pizza store, he notices that
a 14 inch circular pizza has a price of $7 and a 16 inch circular
pizza has a price of $12. He complains the prices are unfair?
Why? What should be the "fair" price of the 16 inch
pizza?
(Source: Adapted from Mathematics Teaching in the Middle School,
Feb 1998)
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The Solution
There are two ways to think about this problem, first in terms
of the diameter of the pizza, and second in terms of the total
area of the pizza.
1) Diameter
We can consider the diameter of the pizza as a means of comparing
price value. A 14 inch pizza that costs $7 would imply a cost
of $0.50 per inch across the diameter if we divide the diameter
by the price. That would lead a consumer to expect a 16 inch pizza
to cost only $8 instead of the $12 charged in the given problem.
In this case the 16 inch pizza works out to be $0.75 per inch
of diameter.
2) Area
We can also conisder the area of a pizza as a means of determining
a "fair" price. The area of a cirlce is found using
the formula A=pi*r^2
For the 14 inch pizza the total area of the pizza would be
A= pi*7^2 = 159.3 square inches. We can take the price of the
pizza and divide it by the total area to determine the cost per
square inch of pizza.
14/159.3 = $0.045 per square inch of pizza.
Using the same method to determine the price per square inch
of the 16 inch pizza produces the following price per square inch
of pizza:
A= pi*8^2 = 201.06 square inches
16/201.06 = $0.06 per square inch of pizza.
Thus we can see that the smaller pizza is cheaper, we can
multiply the cost per square inch of the small pizza by the area
of the larger pizza to determine a "fair" price for
the 16 inch pizza:
$0.045 * 201.06 = $9.05