A client has hired you to build a receptacle in the shape of a rectangle with two half-circles attached at either end. They would like to maximize the area of the rectangular region inside the receptacle. The client has 500 feet of material for you to build with, and they stipulate that the diameter of the two half circles on the ends must be at least 10 feet.
If you use the material only on the outer boundary of the receptacle (i.e., you do not build the two sides of the rectangle that are inside the receptacle), what is the maximum rectangular area you can enclose? What should the dimensions be?