EXAMPLE 2.6.8

Suppose that it costs $480 to build a rectangular wooden deck that measures 6 feet by 8 feet. Assuming that the cost of such an object depends upon its size (area), how much would it cost to build a similar deck measuring 18 feet by 24 feet?

A. $640

B. $1440

C. $1,920

D. $4,320

SOLUTION

It is useful, in a problem involving two similar geometric figures, to compare their sizes via division. In this case we will divide the size (area) of the larger rectangle by the size of the smaller rectangle.
Size of larger deck = Length times Width = (18 feet) times (24 feet) = 432 square feet.
Size of larger deck = Length times Width = (6 feet) times (8 feet) = 48 square feet.
Now, divide:
(area of larger deck) divided by (area of smaller deck)
= 432 divided by 48
= 9
This means that the larger deck is nine times as large as the smaller deck, so it will cost nine times as much, so
Cost of larger deck = 9 times ($480) = $4320.

There are other ways to arrive at this correct answer, such as by forming a proportion and solving for the unknown quantity. Any correct apporach, however, will take into account all three of these considerations:
1. The area of the larger deck.
2. The area of the smaller deck.
3. The fact that the smaller deck costs $480.

What follows is an alternative solution to this problem.

SOLUTION

Since the cost of one of these rectangular objects depends upon its area, we can use the fact that a 6 by 8 foot deck costs $480 to find the cost per square foot of one of these decks. Then we will use the cost per square foot to find the cost of the 18 by 24 foot deck.

To find "cost per square foot" we divide cost by square feet.

The area of the 6 by 8 foot deck is

(6 feet) times (8 feet) = 48 square feet, so

The area of the larger deck is:

(18 feet) times (24 feet) = 432 square feet, so the cost is

(432 square feet) times ($10 per square foot) = $4,320