Area is equal to length times width. The perimeter (the amount of rope) has to equal twice the length added to twice the width so we're left with: A = l * w 200 = 2l + 2w solve for either l or w l = 100 - w plug into the area equation to get one equation with two variables A = w(100 - w) A = -w^2 + 100w take the derivative A' = -2w + 100 set the derivative equal to zero 0 = -2w + 100 2w = 100 w = 50 This is the width that maximizes the area with a width of 50, the length must also be 50 to have a perimeter of 200 therefore, they can rope up to 50 * 50 = 2500 ft^2
