Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.

Verified step by step guidance

1. Identify the variables: Let r be the radius and h be the height of the cylinder. The volume of the cylinder is given by the formula V = πr²h.
2. Set up the volume constraint: Since the volume V is given as 50 m³, we can express this as πr²h = 50, which allows us to express h in terms of r: h = 50/(πr²).
3. Determine the surface area to be minimized: The surface area A to be painted consists of the lateral area of the cylinder and the area of the top. The formula for the surface area is A = 2πrh + πr².
4. Substitute h in the surface area formula: Replace h in the surface area equation with the expression found from the volume constraint to get A in terms of r only: A = 2πr(50/(πr²)) + πr².
5. Differentiate the surface area function A with respect to r, set the derivative equal to zero, and solve for r to find the critical points that minimize the surface area.