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29 November, 22:16

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?

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  1. 29 November, 23:57
    0
    rug area = 108 ft^2

    ...

    let width be x of the strip

    ...

    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.
  2. 30 November, 00:41
    0
    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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