What Are the Optimal Dimensions for a Tank to Minimize Metal Usage?

[Renzokuken]

1.A petrochemical company is designing a cylindrical tank with hemispherical ends to be used in transporting its products. If the volume of the tank is to be 10,000 cubic meters what dimensions should be used to minimize the amount of metal required?



2. V=pi*r^2 + 4/3*pi*r^3
SA= 4*pi*r^2+2*pi*r*h



3. 10000=pi*r^2*h+4/3*pi*r^3
solved for h=-4(pi*r^3-7500)/(3*pi*r^2)
pluged h into SA and then took the partial derivative = 8(pi*r^3-7500)/(3r^2)
r=13.36
Then i pluged r into V equation to solve for h, but h=0 and i don't think it is supposed to

 

[dynamicsolo]

V=pi*r^2 *h + 4/3*pi*r^3
SA= 4*pi*r^2+2*pi*r*h

The fact that you get h = 0 tells you something: the figure with the smallest possible surface area for a given volume is a sphere. (The sphere is one example of what are called "minimal surfaces".) This is related to why soap bubbles are round. Since the problem posed no constraints requiring there to be a cylindrical section for the tank, h = 0 will be the correct result. (In fact, many countries use spherical tanks, appropriately supported, to store natural gas, etc.)