Cynthia Besch wants to buy a rug for a room that is 20 ft wide and 26 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 432 square feet of carpeting. What dimensions should the rug have?

rug area = 108 ft^2

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let width be x of the strip

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width of rug = 12-2x

length of rug = 15-2x

Area of rug = (12-2x) (15-2x) = 108

180-24x-30x+4x^2=108

4x^2-54x+72=0

/2

2x^2-27x+36=0

2x^2-24x-3x+36=0

2x (x-12) - 3 (x-12) = 0

(x-12) (2x-3) = 0

x = 12 OR = x=1.5 feet.

The width will be 1.5 feet.

Let x represent the width of the "uniform strip."

Write formulas: one for the length of the rug, one for the width.

If the room is 20 ft wide, the rug width would have to be 20-2x, and the run length would be 26-2x.

Rug area is 432 sq ft, and this equals (rug length) (rug width), or

432 = (26-2x) (20-2x) = 520 - 40x - 52x + 4x^2 and this is 432.

Subtr. 432 from both sides: 520 - 432 - 92x + 4x^2 = 0. This is a quadratic equation that could be solved in various ways.

4x^2 - 92x + 88 = 0. Let's reduce this by div. all terms by 4:

x^2 - 23x + 22 = 0

Easily factored! Note that - 1x and - 22x add up to - 23x, and that (-1) (-12) = 22. Thus, the factors are (x-1) (x-22) = 0, so that x = 1 or x = 22.

Remembering that x represents the strip width, we eliminate x = 22 and keep x = 1.

The rug dimensions should be (26-2) by (20-2), or 24 by 18 feet.

Check: Does this area come out to 432 sq ft? (24) (18) = 432 / YES