SOLUTION: Cynthia Besch wants to buy a rug for a room that is 18 feet wide and 19 feet long. She wants to leave a uniform strip around the rug. She can afford 272 square feet of carpeting. W

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: Cynthia Besch wants to buy a rug for a room that is 18 feet wide and 19 feet long. She wants to leave a uniform strip around the rug. She can afford 272 square feet of carpeting. W      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 992571: Cynthia Besch wants to buy a rug for a room that is 18 feet wide and 19 feet long. She wants to leave a uniform strip around the rug. She can afford 272 square feet of carpeting. What dimensions should the rug have?
272+(19-2x)(18+2x)
1, 35/2
But, what are the dimensions in feet? How do I convert? Thanks.
Found 2 solutions by rothauserc, vleith:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
The border will be inside the dimensions of the room, therefore
(19-2x) * (18-2x) = 272
342 - 74x + 4x^2 = 272
rearrange the terms
4x^2 - 74x + 342 = 272
divide both sides of = by 4
x^2 - 18.5x + 85.5 = 68
subtract 68 from both sides of =
x^2 - 18.5x + 17.5 = 0
use quadratic formula to solve for x
x = (-(-18.5) + sqrt((18.5^2)−(4×17.5))) / 2 = 17.5
x = (-(-18.5) - sqrt((18.5^2)−(4×17.5))) / 2 = 1
****************************************************************
The reasonable dimension for the strip around the rug is 1 foot, therefore
the dimensions for the rug are 17 feet long and 16 feet wide

Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Let the width of the strip be x
The rug will have square footage of (18-2x)*(19-2x)
you are told she wants a rug that is 272 sq ft
so 272+=+%2818-2x%29%2A%2819-2x%29
272+=+342+-+36x+-+38x+%2B+4x%5E2
0+=+4x%5E2+-+74x+%2B+70
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-74x%2B70+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-74%29%5E2-4%2A4%2A70=4356.

Discriminant d=4356 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--74%2B-sqrt%28+4356+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-74%29%2Bsqrt%28+4356+%29%29%2F2%5C4+=+17.5
x%5B2%5D+=+%28-%28-74%29-sqrt%28+4356+%29%29%2F2%5C4+=+1

Quadratic expression 4x%5E2%2B-74x%2B70 can be factored:
4x%5E2%2B-74x%2B70+=+4%28x-17.5%29%2A%28x-1%29
Again, the answer is: 17.5, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-74%2Ax%2B70+%29


so the strip will be 1 ft wide
That makes the rug (18-2) by (19-2) feet
16 feet by 17 feet