Question 1027377: Cynthia Besch wants to buy a rug for a room that is 17 ft wide and 24 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 144 square feet of carpeting. What dimensions should the rug have?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the size of the room is 17 feet in width and 24 feet in length.
she wants to leave a uniform width of floor around the rug.
if we let that width be equal to the variable named x, then we get:
the dimensions of the room are 17 * 24.
the dimensions of the rug need to be (17-2x) * (24 - 2x).
this will leave a width of x surrounding each width of the rug and each length of the rug.
since the area of the rug is equal to 144, then we get:
(17-2x) * (24-2x) = 144.
simplify this equation by multiplying the expression on the left side using the distributive property of multiplication to get:
17*24 - 34x - 48x + 4x^2 = 144
simplify and combine like terms to get:
408 - 82x + 4x^2 = 144
rearrange the terms in descending order of degree to get:
4x^2 - 82x + 408 = 144
subtract 144 from both sides of the equation to get:
4x^2 - 82x + 264 = 0
take out the greatest common factor of 2 to get:
2 * (2x^2 - 41x + 132) = 0
factor the quadratic equation, using the quadratic formula, to get:
x = 4 or x = 16.5
our original equation is (17-2x) * (24-2x) = 144.
when x = 16.5, the dimensions become 0 which is not valid, so x = 16.5 is an extraneous solution and is discarded.
when x = 4, the dimensions becomes (17-8) * (24-8) = 144.
this results in 9 * 16 = 144 which results in 144 = 144 which confirms this solution is good.
we have x = 4.
the rug will be 9 by 16 and will have a border of 4 feet surrounding each length of the rug and each width of the rug.
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