Minimizing calculus

Question

A cylindrical can is to be made to hold 250 π cm3.Find the dimensions of the can that will minimize the cost of the metal to manufacture it.

Answer 1

Let r = radius of the base and h = height

Then, πr²h = 250π. So, h = 250 / r².

Assuming that the can has a top, we are to minimize the surface area, S, where

S = 2πr² + 2πrh = 2πr² + 500π / r, r > 0

dS/dr = 4πr - 500π / r² = (4πr³ - 500π) / r² = 0 when r³ = 125. So, r = 5.

When 0 < r < 5, dS/dr < 0. So, S is decreasing.

When r > 5, dS/dr > 0. So, S is increasing.

We see then, from the shape of the graph of the surface area function, that S is minimized when r = 5 cm and h = 10 cm.