Question

Cynthia Besch wants to buy a rug for a room that is 19 ft wide and 32 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 420 square feet of carpeting. What dimensions should the rug have?

Answer 1

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 $l$ by $h$, the width of the strip of floor is $w$.

We are told that Cynthia can only afford 420 square feet of carpeting; therefore, it must be that

$l*h=420$ (this says the area of the rug must be 420 square feet)

From the figure we see that

$l= 32-2w$

$h=19-2w$

Therefore,

$l*h=420\\\\(32-2w)(19-2w)=420$

We expand this equation and get:

$4w^2-102w+608=420\\\\4w^2-102w+188=0$

using the quadratic equation we get two solutions:

$w=2\\w=23.5$

since the second solution, namely $w=23.5$, 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 $w=2$ to be our solution.

Now we find the dimensions of the rug:

$l=32-2(2)=28\\\\h=19-2(2)=15$

The rug is 15 ft wide and 28 ft long.