Answer
The dimensions of the rug are 9ft (width) by 12ft (length).
Work Step by Step
There is 72 square feet available after the carpet is laid down.
$$72=24x+2x(15-2x)$$
$$72=24x+30x-4x^2$$
$$4x^2-54x+72=0$$
$$x^2-27x+36=0$$
$$x=12,1.5$$
x=1.5, because one of the dimensions of the room is 12 feet.
$$12-2(1.5)=9$$
$$15-2(1.5)=12$$
The dimensions of the rug are 9ft (width) by 12ft (length).