The formula for the perimeter, P, of a rectangle is P = 2(l + w), where l represents the length and w represents the width.
A. Todd has a string that is 26 inches long and wants to create a 9-inch-long rectangle with it. Substitute these values into the formula above, and solve for w to determine how wide Todd’s rectangle must be. Show your work.
B. Todd solved for the width arithmetically, as follows:
The length of the rectangle is 9 inches, so the sum of the lengths of the two long sides is 2 • 9 = 18 inches.
The string is 26 inches long, and 18 inches are used for the long sides of the rectangle, so the number of inches available for the short sides is 26 − 18 = 8 inches.
A rectangle has 2 congruent short sides, so the length of each short side, or the width of the rectangle, is 8 ÷ 2 = 4 inches.
Describe how Todd’s arithmetic solution process is similar to your algebraic solution process from part A, and how they are different.
C. The perimeter formula can be solved for w, in terms of P and l, as shown below.
P = 2(l + w)
P = 2l + 2w
P − 2l = 2w
= w
Explain why it could be useful to solve the perimeter equation for w in terms of P and l.