Question

Find the minimum cost of producing a can that contains 1000mm^3 if the cost of the metals is 2 cents per mm^2 and the cost to weld the seams is 1 cent per mm. Assume three welds;one around the top, one around the bottom, and one down the side.

Answer 1

let the radius of the can be r mm
and its height be h mm

area = 1 rectangle + 2 circles
= 2pi(r)(h) + 2pi(r^2)


seems length = 2 circuferences + height of can
= 2(2pi(r)) + h
= 4pi(r) + h

so Cost = 2[2pi(r)(h) + 2pi(r^2)] + 1[4pi(r) + h]

but pi(r^2)h = 1000
so h = 1000/(pi(r^2))

this should give you a good start, sub h back into the Cost equation,
find the derivative of Cost,
then solve for r

sub back into Cost equation to find minimum cost