Question 1198424
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Answer: <font color=red>20 ft by 28 ft</font>


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Explanation:


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


Diagram
*[illustration Screenshot_146.png]
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 = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-104)+-sqrt((-104)^2-4(4)(100)))/(2(4))}}}


{{{x = (104+-sqrt(10816-1600))/(8)}}}


{{{x = (104+-sqrt(9216))/(8)}}}


{{{x = (104+-  96)/(8)}}}


{{{x = (104+96)/(8)}}} or {{{x = (104-96)/(8)}}}


{{{x = (200)/(8)}}} or  {{{x = (8)/(8)}}}


{{{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.
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