a rectagular sheet of metal has perimeter 36 meters. maximize the volume if the sheet is rolled into a cylinder
if the sheet is x by y,
2x+2y=36
x = 18-y
rolled parallel to x,
v = pi r^2 x
where r = y/2pi
v = pi (y/2pi)^2 (18-y)
= y/4pi (18-y)
dv/dy = (9-y)/2pi
dv/dy = 0 when y=9
as expected, it's a 9x9 square sheet.