The most of the expense is incurred in joining the sides to the rims of the can. If we cut the discs from hexagons then the total cost is proportional to

4√3 r2 + 2pi*r*h + k(pi* r + h)

where k is the reciprocal of the length that can be joined for the cost of one unit area of metal. show that this expression is minimized when

(V^1/3)/k= ((pih)/r)^1/3 * (2pi-h/r)/(pih/r-4*3^(1/3))