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  by
 by  , the width of the strip of floor is
, the width of the strip of floor is  .
.
 We are told that Cynthia can only afford 420 square feet of carpeting; therefore, it must be that 
 <em>(this says the area of the rug must be 420 square feet)</em>
 <em>(this says the area of the rug must be 420 square feet)</em>
From the figure we see that


Therefore, 

We expand this equation and get:

using the quadratic equation we get two solutions:

since the second solution, namely  , 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
, 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  to be our solution.
 to be our solution.
Now we find the dimensions of the rug:

The rug is 15 ft wide and 28 ft long.