I cannot seem to get the last two steps to get the correct answer.

Question

Cynthia Besch wants to buy a rug for a room that is 20 ft wide and 31 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 432 square feet of carpeting.What dimensions should the rug have?

Answer 1

I'm a fan of the guess-and-check method. The dimensions of 31 (length) and 20 (width) are 11 apart. We know that length times width equals area. So what length times what width will give us 432 and be 11 apart? I try dividing 432 by 12 and get 36. More than 11 apart. I try dividing 432 by 16 and get 27. That's 11 apart -- AND the same less for each of the room's dimensions! So the rug will be 27 feet long by 16 feet wide.

Answer 2

Drawing a picture will help. Start with a rectangle 20x31 which represents the room. Draw a second rectangle inside this representing the rug. Call the distance from the rug to the wall x, then the dimensiions of the rug will be (20-2x) x (31-2x) so (20-2x)(31-2x)=432

4x^2 -102x +620 = 432

4x^2 -102x +188 =0

2x^2 -51x +94 =0

(2x-47)(x-2) = 0 so solutions are x=2 and x=47/2

You need to discard 47/2 since this would give you negative distances so for x=2 the dimensions are 16x27

Fun problem

Answer 3

Given there is a uniform strip around the rug, we will call that dimension x.

the area of the rug will be (20ft - x)(31ft - x)= 432 ft^2

first we solve for x.

(620 - 51x + x^2) = 432

x^2 -51x +188 = 0

x = 4ft

20 - 4 = 16ft

31 - 4 = 27ft

the rug should have dimensions of 16ft by 27ft

to check 16*27=432