The dimensions the rug should have is 20 by 29
The dimension of the rug is given as:
22 ft by 31 ft
The area of the carpet is given as:
Area = 580
Represent the length of the empty space with x.
So, the dimension of the carpet is:
22 - 2x by 31 - 2x
The area is calculated as:
Area = (22 - 2x) * (31 - 2x)
Expand
Area = 682 - 62x - 44x + 4x^2
Evaluate the difference
Area = 682 -106x + 4x^2
Substitute Area = 580
682 -106x + 4x^2 = 580
Subtract 580 from both sides
102 -106x + 4x^2 = 0
Rewrite as:
4x^2 - 106x +102 = 0
Using a graphing calculator, we have:
x = 1 and x = 22.5
Substitute these values in 22 - 2x by 31 - 2x
So, we have:
Length = 22 - 2(1) = 20 or Length = 22 - 2(22.5) = -23
Width = 31 - 2(1) = 29 or Length = 31 - 2(22.5) = -14
The dimensions cannot be negative.
So, we have:
Length = 20 and Width = 29
Hence, the dimensions the rug should have is 20 by 29
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