Answer:
The rug should be 15 ft wide and 28 ft long.
Step-by-step explanation:
I have attached a figure that represents the situation.
The the rug is [tex]l[/tex] by [tex]h[/tex], the width of the strip of floor is [tex]w[/tex].
We are told that Cynthia can only afford 420 square feet of carpeting; therefore, it must be that
[tex]l*h=420[/tex] (this says the area of the rug must be 420 square feet)
From the figure we see that
[tex]l= 32-2w[/tex]
[tex]h=19-2w[/tex]
Therefore,
[tex]l*h=420\\\\(32-2w)(19-2w)=420[/tex]
We expand this equation and get:
[tex]4w^2-102w+608=420\\\\4w^2-102w+188=0[/tex]
using the quadratic equation we get two solutions:
[tex]w=2\\w=23.5[/tex]
since the second solution, namely [tex]w=23.5[/tex], is larger than one of the dimensions of the room (is greater than 19 ft) it cannot be the width of the strip; therefore, we take [tex]w=2[/tex] to be our solution.
Now we find the dimensions of the rug:
[tex]l=32-2(2)=28\\\\h=19-2(2)=15[/tex]
The rug is 15 ft wide and 28 ft long.
