A cylindrical can is to be made to hold 1 liter of oil. Find the radius that will minimize the cost of the metal to manufacture the can, assuming the cost of the metal is proportional to the surface area.

Core Concept

Let r be the radius and h be the height of the cylinder. The volume is V = πr²h = 1000 cm³ (1 liter = 1000 cm³). The surface area is A = 2πr² + 2πrh. From the volume equation, h = 1000/(πr²). Substitute into A: A = 2πr² + 2πr(1000/(πr²)) = 2πr² + 2000/r. Take the derivative with respect to r: A' = 4πr - 2000/r². Set A' = 0: 4πr - 2000/r² = 0, which gives 4πr = 2000/r², so 4πr³ = 2000, and r³ = 500/π. Taking the cube root gives r = (500/π)^(1/3) = √(1000/π)^(1/3) = √π cm (approximately 5.4 cm).