Question 1027377
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.