EXAMPLE 2.6.9

A circular pizza that is 16 inches in diameter costs $12.00. Assuming that the cost of such a pizza depends upon its size (area), what would be the cost of a pizza that is 8 inches in diameter?

A. $3.00

B. $4.00

C. $6.00

D. $8.00

 

SOLUTION

We will compare the size (area) of the large circular pizza with the size (area) of the smaller circular pizza, by division.
The diameter of the large pizza is 16 inches, so its radius is 8 inches.
The diameter of the small pizza is 8 inches, so its radius is 4 inches.

Area of large pizza = pi times (r squared) = pi times (8 squared) = 64 pi square inches.
Area of small pizza = pi times (r squared) = pi times (4 squared) = 16 pi square inches.
Now, divide:

(Area of large pizza) divided by (Area of small pizza)
= (64 pi square inches) divided by (16 pi square inches)
= 4

This means that the large pizza is four times as large as the small pizza, so the large pizza costs four times as much as the small pizza. Thus, to find the cost of the smaller pizza, we divide the cost of the larger pizza by 4:

$16 divided by 4 = $4. The correct choice is B.

There are other ways to arrive at this correct answer, such as by forming a proportion and solving for the unknown quantity. Any correct solution, however, will take into account the area of the larger pizza, the area of the smaller pizza, and the fact that the larger pizza costs $16. What follows is another correct way to solve this problem.

SOLUTION

We can use the fact that the 16-inch diameter pizza costs $12.00 to find the cost per square inch of one of these pizzas. Then we will use the cost per square inch to find the cost of the other pizza.

To find cost per square inch, we divide cost by square inches. The $12.00 pizza has a diameter of 16 inches, so it has a radius of 8 inches. Its area is computed as follows:

Area = pi times (radius squared) = pi times (8 squared) = 201.06 square inches

Now we find the cost per square inch:

cost per square inch = $12 divided by 201.06 square inches = $0.0597 per square inch

The other pizza has a diameter of 8 inches, so it has a radius of 4 inches; thus the area of the other pizza is computed as follows:

Area = pi times (radius squared) = pi times (4 squared) = 50.27 square inches

Finally, we multiply to find the cost:

Cost = (50.27 square inches) times ($0.0597 per square inche) = $3.00

The correct choice is A. Notice the the "obvious" choice, C, was incorrect. This is because a circle with a 16-inch diameter is fours times as big as circle with an 8-inch diameter (not twice as big).