An 800 cubic meter cylindrical container made of sheet metal, including top and bottom is to be designed.?

What dimensions for h(height) and r (radius) will use the least material? Use calculus methodAn 800 cubic meter cylindrical container made of sheet metal, including top and bottom is to be designed.?
The top and bottom each use 蟺 * r^2 of material


The cylindrical portion uses 2 * 蟺 * r * h of material





Together,





A = 2 * 蟺 * r^2 + 2 * 蟺 * r * h of material





Also,





800 = 蟺 * r^2 * h


h = 800 / (蟺 * r^2)





A = 2 * 蟺 * r^2 + 2 * 蟺 * r * 800 / (蟺 * r^2)


A = 2 * 蟺 * r^2 + 1600 / r





Take the first derivative of A:





A' = 4 * 蟺 * r - 1600 / r^2 = 0


0 = 蟺 * r - 400 / r^2


0 = 蟺 * r^3 - 400


400/蟺 = r^3


r = cube root(400/蟺)