I can't even explain how lost i am on this problem so here it is.
An indoor physical fitness room consists of a rectangular region with a semicircle on each end. If the perimeter of the room is to be a 200-meter running track, find the dimensions that will make the are of the rectangular region as large as possible.
You want to maximize the area of a rectangular region, represented by , with the constraint of a 200 meter perimeter. The perimeter is going to be equal to the circumference of a circle (with diameter x) plus two sides (of length y) of the rectangle:
So given that maximize . Then I would assume you would use Lagrange multipliers.