Calculus Question (1)?

Question

A right circular cylinder is to be designed to hold 20 cubic inches of soft drink. The cost of the material for the top and bottom of the can is twice the cost of the material of the sides. Let r represent the radius and h the height of the cylinder. a.) Write the equation for the surface area SA in terms or r and h. b.) Write the cost function C. c.) Write the cost function as a function of one variable, r. d.) Find the radius that minimizes cost. Thank you, I'm really confused on these minimization/ max problems. Please explain each step.

Answer 1

S = 2 pi r^2 +2pi r*h
C = 4 pi r^2 *x +2pir h*x where x is the unit cost of the material
which is a constant
V= pi*r^2*h =20 so h= 20/pi r^2

C= 2 pi x (2r^2+ 20/pi r)
Extremes of
K = r^2+10/pi *r
K´= 2r -10/ pir^2 = 0
r^3-5/pi=0 so r = (5/pi)^1/3
sign of K´ --------(5/pi)^1/3 ++++++ so it´s a minimum
Check calculations