a rectagular sheet of metal has perimeter 36 meters. maximize the volume if the sheet is rolled into a cylinder

Answer 1

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.