Question

Find the minimum amount of sheet that can be made into a closed clinder having a volume of 120 cu inches.

Answer 1

let the radius of the cylinder be r inches
and its height be h inches
given: V = 120 inches^3
or
π r^2 h inches^3 = 120 inches^3
h = 120/(πr^2)

Surface area
= 2πr^2 + 2πrh
= 2πr^2 + 2πr(120/πr^2)
= 2πr^2 + 240/r
d(surface area)/dr = 4πr - 240/r^2
= 0 for a min of surface area

4πr = 240/r^3
r^3 = 60/π
r = (60/π)^(1/3) = appr 2.673 inches
sub that back into
2πr^2 + 240/r to find the min surface area

Answer 2

A = 2 pi r h + 2 *pi r^2
V = 120 = pi r^2 h
so h = 120/ (pi r^2)

A = 2 pi r 120/ (pi r^2) + 2 pi r^2
A = 240 / r + 2 pi r^2

dA/dr = -240/r^2 + 4 pi r
= 0 for max or min
4 pi r = 240/r^2
r^3 = 60 / pi
do not trust my arithmetic, working fast