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Explanation:
Refer to the diagram below.
The outer dimensions 24 ft and 33 ft shrink down to 24-2x ft and 33-2x ft respectively. This subtraction of 2x is due to subtracting two copies of x per side.
The carpet has area of (24-2x)(33-2x)
Cynthia can afford to buy 400 sq ft of carpet
So we set 400 equal to that previous expression and solve for x
(24-2x)(33-2x) = 400
24(33-2x) - 2x(33-2x) = 400
792-48x - 66x + 4x^2 = 400
4x^2 - 114x + 792 = 400
4x^2 - 114x + 792-400 = 0
4x^2 - 114x + 392 = 0
From here, use the quadratic formula to isolate x.
Plug in a = 4, b = -114, c = 392
The two possible solutions are x = 24.5 and x = 4
But if x = 24.5, then 24 - 2x = 24 - 2*24.5 = -25 which isn't possible. We cannot have a negative width or negative length for the carpet.
Luckily x = 4 does work since
24 - 2x = 24 - 2*4 = 16
33 - 2x = 33 - 2*4 = 25
Both results are positive.
Therefore, the carpet has dimensions of <u>16 ft by 25 ft</u>
Check: 16*25 = 400, so the answer is confirmed
