if the sheet is used to make an enclosed cylinder, with a top, bottom and side then the following is true 9x9 cm:
but if open top& bottom, volume maximizing dimensions are 12 by 6 cm
first consider the closed top& bottom cylinder:
36/4 = 9 cm by 9 cm sheet gives the greatest Volume
as a square sheet gives maximum area and max surface area gives max Volume
no need to do further calculations assuming closed top&bottom
but here are some:
Volume of a cylinder = hA= hpir^2
A= Area of the cylinder's base
Surface Area=SA= pidh+2A = 2pirh + 2pir^2 =2pir(h+r)
SA= 36=2pi(hr +r^2)
18/pi = hr + r^2
hr = 18/pi -r^2
h = 18/pir - r
V= hpir^2 = (18/pir -r)pir^2
=18r - pir^3
max V is when V'=0
V'= 18- 3pir^2=0
pir^2 =18/3 = 6
r^2= 6/pi
r= sqr(6/pi)
h= 18/pir-r = 18/pi(sqr(6/pi) - sqr(6/pi)
max V= hpir^2= hpi(6/pi) = 6h = 16.58 cm^3
but if the cylinder is made from the sheet with open top & bottom then the answer is different, with a much larger maximum volume:
the sheet = hx where h= height of the cylinder = one dimension of the sheet, and x = the other dimension of the sheet = circumference of the base of the cylinder = x= pid = 2pir. radius of the cylinder = r = x/2pi.
Volume = V= hA = hpir^2 =hpi(x/2pi)^2 = hx^2/4pi cm^3
36=Perimeter= 2h+2x
18= h+x
h=18-x
V= (18-x)(x^2/4pi) = 9x^2/2pi - x^3/4pi
max V is when V'=0
v' = 9x/pi -3x^2/4pi = 0
9x -3x^2/4 = 0
(9-3x/4)=0
x=9(4/3)=12 cm
h=18-12= 6 cm
then max V = hpir^2= 6(pi)(x/2pi)^2 = 6pi(12/2pi)^2= 6pi(36/pi^2) = 216/pi
= about 216/3.141593 = 68.755 cm^3
