The width of a rectangle is 3 ft less than the length. The area of the rectangle is 180 ft2. Find the length L and width W of the rectangle.

Question

The width of a rectangle is 3 ft less than the length. The area of the rectangle is 180 ft². Find the length L and width W of the rectangle.

Answer 1

Here's what we know

A = Lw (Rectangle are equals length * width)

w = L - 3 (width is 3 feet less than length)

A = 180 (rectangle area equals 180 ft squared)

That gives us:

180 = L(L - 3)

Distribute:

L² - 3L = 180

Subtract 180:

L² - 3L - 180 = 0

Now we have a quadratic to work with. We need to find two numbers that total up to -3, and multiply to -180. That we have a -3 means that these absolute values are going to be rather close.

180 ÷ 10 = 18. Approaching, but not quite close enough. Gives a good baseline, though.

180 ÷ 12 = 15 There we go. 12 - 15 = -3

So we now have the following factorization:

(L + 12)(L - 15) = 0

L + 12 = 0 and L - 15 = 0

L = -12 and L = 15

We cannot have a negative length for a side, so L = 15 ft. And w = 15 - 3, which is 12 ft.