I has a rectangular piece of fabric that has an area of 200 in and a perimeter of 66 in. What is the length and width of this fabric. Show or explain your work.

Answer 1

Here's what we know:

200 = Lw (length * width = area, which is 200 in²)

66 = 2L + 2w (2 lengths + 2 widths = perimeter, which is 66)

So what we need to do is isolate a variable in the latter equation to place in the former. Start by dividing by 2, then subtract one of the variables to the other side:

33 = L + w

33 - w = L

Now that we have a value for a variable, we place it into the other equation:

200 = w(33 - w)

w² + 33w = 200

-w² + 33w - 200 = 0

By subtracting 200 from both sides, we now have a quadratic to factor. Multiply by -1 to get rid of that negative in front of the square:

w² - 33w + 200 = 0

That +200 along with a -33 means that the factoring is going to have a subtractions in both factors:

(w - m)(w - n) = 0

Now to find m and n, we need two numbers that multiply together to make 200 and total a negative 33:

200 ÷ 8 = 25

-25 - 8 = -33

So that means we have:

(w - 25)(w - 8) = 0

w - 25 = 0; w - 8 = 0

w = 25, 8

Check through using the perimeter equation:

66 = 2 * 25 + 2 * 8

66 = 50 + 16

66 = 66