Question

A closed aluminum can (right circular cylinder with top and bottom) is required to hold 128π cubic inches of liquid. Find the radius and the height of the can that require the least amount of aluminum. A closed aluminum can (right circular cylinder with top and bottom) is required to hold 128π cubic inches of liquid. Find the radius and the height of the can that require the least amount of aluminum. @Mathematics

Answer 1

you'll want to minimize the surface area: S = 2pi*r*h + pi*r^2 with the restriction that
V=128=pi*r^2*h

Answer 2

solve for h and replace into the S formula...

Answer 3

V = πhr^2
πhr^2 = 128π
hr^2 = 128
h = 128/(r^2)
A = 2πrh + 2πr^2
A = 2πr(128/(r^2)) + 2πr^2
A = 256π/r + 2πr^2
dA/dr = 4πr - 256π/(r^2)
For min amt of Al:
dA/dr = 0
4πr^3 - 256π = 0
r^3 = 64
r = 4
When r = 4,
h = 128/(4^2) = 8
Height = 8 inches, Radius = 4 inches.