Question 1113179
let the distance between the rug and the walls be x on all sides.


the area of the room is 30 * 23 = 690


the area of the rug is (30 - 2x) * (23 - 2x) = 408


perform the multiplication to get:


690 - 60x - 46x + 4x^2 = 408


combine like terms to get:


690 - 106x + 4x^2 = 408


subtract 408 from both sides of the equation to get:


690 - 106x + 4x^2 - 408 = 0


combine like terms to get:


282 - 106x + 4x^2 = 0


reorder the terms in descending order of degree to get:


4x^2 - 106x + 282 = 0


factor this quadratic equation to get:


x = 23.5 or x = 3


x can't be 23.5 because then the area of the rug would not be equal to 408 because:


(30 - 2 * 23.5) * (23 - 2 * 23.5) equals a negative.


x has to be 3 or none of the possible solutions are good.


when x = 3, you get (30 - 6) * (23 - 6) = 408.


this results in 24 * 17 = 408 which becomes 408 = 408 which is true.


your solution is that x = 3 and the dimensions of the rug are 24 by 17.


this rug fits into a room that is 30 by 23.


there will be a border around the rug that is 3 feet in all directions.


the dimensions of the room are 30 by 23


the dimensions of the rug are 24 by 17.


the difference between the length of the room and the length of the rug is 30 - 24 = 6.


the difference between the width of the room and the width of the rug is 23 17 = 6.


divide this difference by 2 and you get 3 foot border between the rug and the wall all around.


here's my picture.


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