A 400 meter track enclosed a field which consists of a recta

A 400 meter track enclosed a field which consists of a rectangle (shaded), and 2 semi-circles. Find the dimensions of the rectangle, x and y, so that the area of the rectangle a maximum. Note: The 400 meters is the measurement of the 2 \"y\" sides and the 2 semi-circles. It does not include the 2 \"x\" sidest (Circumference of a circle = 2 pi r)

Solution

2r=x total length= 2pir+2y=400 ==>xpi+2y=400 area=x*y ==>x(100-pix)/2=50x-pix^2 /2 maximum area==>derivative=0 ==>50-pix=0 ==>x=50/pi ==>y=175