Question

A rectangular piece of paper with perimeter
100 cm is to be rolled to form a cylindrical
tube. Find the dimensions of the paper that
will produce a tube with maximum volume.

I have made it up to getting an equation for V(w).

Answer 1

if the sheet is x by y, and is rolled along the y axis,

2x+2y = 100
v = pi r^2 y
where 2pi r = x, or r = x/(2pi), so
v = pi (x/(2pi))^2 (100-2x)/2
= x^2(50-x)/(4pi)

dv/dx = x(100-3x)/(4pi)
dv/dx=0 when x = 100/3

at that point, max v is pi*(50/3)^3