EXAMPLE 2.6.10

The morning after a party, Gomer finds on his living room carpet a circular purple wine stain with a diameter of 1 foot. Homer's Carpet Service charges him $30 to remove the stain. Assuming that the cost of removing such a stain depends upon its size, how much would it cost to remove a stain that 18 inches in diameter?

A. $45 .00

B. $67.50

C. $54.00

D. $90.00

SOLUTION

We will compare the size (area) of the large circular stain with the size (area) of the smaller circular stain, by division.
The diameter of the large stain is 18 inches, so its radius is 9 inches.

The diameter of the small stain is 1 foot, or 12 inches, so its radius is 6 inches.
Area of large stain = pi(r²) = pi(9²) = 81pi square inches.
Area of small stain = pi(r²) = pi(6²) = 36pi square inches.
Now, divide:

(Area of large stain)/(Area of small stain)
= (81pi square inches)/(36pi square inches)
= 2.25

This means that the large stain is 2.25 times as large as the small stain, so removal of the large stain costs 2.25 times as much as removal of the small stain. Thus, to find the cost of removal of the larger stain, we multiply the cost of the smaller stain by 2.25:

(2.25)($30) = $67.50. 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 stain, the area of the smaller stain, and the fact that removal of the smaller stain costs $30. What follows is another correct way to solve this problem.

We can use the fact that removal of the 1-foot diameter stain costs $30 to find the cost per square foot of stain removal. Then we will use the cost per square foot to find the cost of removing the other stain.

To find cost per square foot, we divide cost by square feet. The $30 stain has a diameter of 1 foot, so it has a radius of .5 feet. Its area is computed as follows:

Now we find the cost per square foot:

The other stain has a diameter of 1.5 feet (18 inches), so it has a radius of .75 feet; thus the area of the other stain is computed as follows:

Finally, we multiply to find the cost:

Cost = (1.77 sq. ft.)($38.22 per sq. ft.) = $67.65

The best choice is B; the slight difference between our answer and the answer in choice B is due to rounding.