Question

Calculus: A pen is to be constructed at the bend in a river using one mile of fence. What should x and y be to maximize the area?

Answer 1

It depends on what shape the pen is to be. If it is going to be a quadrilateral, then its area will be x*y. We want x*y to be maximized and we know that 2x+y must be equal to 1. You could also use x+2y if you wish, we need to show that one side is the river, which will not require fencing. Either way, you will get the same dimensions. We can solve for y if we use 2x+y=1. y=1-2x. We can substitute 1-2x for y in the area equation and get Area=x(1-2x) If we distribute the x and differentiate, we obtain A'=-4x+1. The area will be maximized where the derivative is 0. 0=-4x+1. x=1/4. If 2x+y=1 then .5+y=1 which implies that y=1/2. So x=.25 and y=.5 give the dimensions for maximized area.