SOLUTION: Cynthia Besch wants to buy a rug for a room that is 22 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 560 square feet of

Algebra ->  Square-cubic-other-roots -> SOLUTION: Cynthia Besch wants to buy a rug for a room that is 22 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 560 square feet of      Log On


   

Question 1198424: Cynthia Besch wants to buy a rug for a room that is 22 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 560 square feet of carpeting. What dimensions should the rug​ have?
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 20 ft by 28 ft

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Explanation:

x = width of the uniform strip (i.e. the gap between the rug edge and the wall)

Diagram

The larger outer rectangle is 22 by 30
The smaller inner rectangle inside is (22-2x) by (30-2x)
For each inner dimension, we're subtracting two copies of x from the outer dimension.

The smaller rectangle area is 560 sq ft which is the amount of carpeting she can afford.

length*width = area
(22-2x)*(30-2x) = 560
22*(30-2x) - 2x*(30-2x) = 560
660-44x - 60x + 4x^2 = 560
4x^2 - 104x + 660 = 560
4x^2 - 104x + 660-560 = 0
4x^2 - 104x + 100 = 0

Compare this to ax^2+bx+c = 0
and we see that:
a = 4
b = -104
c = 100

Plug those values into the quadratic formula
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-104%29%2B-sqrt%28%28-104%29%5E2-4%284%29%28100%29%29%29%2F%282%284%29%29

x+=+%28104%2B-sqrt%2810816-1600%29%29%2F%288%29

x+=+%28104%2B-sqrt%289216%29%29%2F%288%29

x+=+%28104%2B-++96%29%2F%288%29

x+=+%28104%2B96%29%2F%288%29 or x+=+%28104-96%29%2F%288%29

x+=+%28200%29%2F%288%29 or x+=+%288%29%2F%288%29

x+=+25 or x+=+1

Those are the potential gap widths from the edge of the carpet to the wall.

We need to check each to see if they would be valid.

If x = 25, then
22-2x = 22-2*25 = -28
which isn't valid. A negative length doesn't make sense.
Therefore we ignore x = 25. It's not a valid solution.

If x = 1, then
22-2x = 22-2*1 = 20
30-2x = 30-2*1 = 28
Both results are positive, so x = 1 is valid.

The carpet has dimensions of 20 ft by 28 ft
Check: 20*28 = 560 sq ft
The answers are confirmed.

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Cynthia Besch wants to buy a rug for a room that is 22 ft wide and 30 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 560 square feet of carpeting. What dimensions should the rug​ have?
Let the width of the uniform strip be W
With dimensions of the room being 22 ft by 30 ft, dimensions of the rug will be 22 - 2W, and 30 - 2W
We then get the following RUG-AREA equation: (22 - 2W)(30 - 2W) = 560 
                                             2(11 - W)2(15 - W) = 2(2)(140) ------ Factoring out 2(2), or 4 on each side
                                               (11 - W)(15 - W) = 140
                                                  
                                                (W - 1)(W - 25) = 0 --- Factoring trinomial
                                                 W - 1 = 0     OR       W - 25 = 0
Width of unifrom strip, or W = 1       or      W = 25 (ignore, as width of strip CANNOT be > the room's width or length)

With width (W) of strip being 1 foot, dimensions of rug are: 22 - 2(1) by 30 - 2(1) = 20 ft by 28 ft.

** NOTE: You don't need to use the quadratic equation formula as the other person did. Too much work, too much time,
         and DEFINITELY too many large numbers, which can make one, prone to errors!!