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?

The dimensions of the rug should be 18 feet and 24 feet

The given parameters;

dimension of the room = 20 ft by 26 ft

maximum area of rug she can afford = 432 ft²

To find:

the dimension of the rug

If they will be a uniform stripe of floor around the rug, then let the uniform excess length of the floor to removed from each dimension = y

The value of y cannot be greater than any of the original dimension, we will choose y = 2

The uniform dimension of the floor to be covered by the maximum area of rug she can afford = (20 - 2) and (26 - 2) = 18 and 24

Thus, the dimensions of the rug should be 18 feet and 24 feet

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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

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