Wendy has 180 feet of fencing. She needs to enclose a rectangular space with an area that is ten times its perimeter. If she uses up all her fencing material, how many feet is the largest side of the enclosure?
I need to find the dimensions of a rectangle that has a perimeter of 180 feet and an area of 10 times that, or 1800 square feet. Let me call the length and width of the rectangle x and y, respectively. Then I have two equations relating x and y: 2x + 2y = 180 and xy = 1800. I can simplify the first equation by dividing both sides by 2, getting x + y = 90. Then I can substitute y = 90 - x into the second equation, getting x(90 - x) = 1800. This is a quadratic equation that I can solve by expanding, rearranging, and factoring. I get x^2 - 90x + 1800 = 0, which factors into (x - 60)(x - 30) = 0. So x is either 60 or 30, and y is either 30 or 60, depending on which one I choose for x. These are the possible dimensions of the rectangle, and the largest side is 60 feet.

