For a closed cylinder with radius r cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 32 cm3

Question

anonymous · Answer

The surface area is the circumference of a circle, progressed along the cylinder's length. So the surface area is given by:
    $$S = 2\pi rh$$
Now, to minimize the surface area, we can take the derivative of surface area and set it equal to 0. Before doing so however, notice that both r and h are variables of s. It would be easier if we had a means of turning it into a single variable. We can do so, by noting that the volume V is constant. The volume of a cylinder is the area of a circle progressed along the cylinder's length. So the volume is given by:
      $$V = \pi r^2 h$$
Solve, for either r or h in terms of the other, and plug this into the surface area equation. 
This will only leave you with S in terms of a single variable. Now take the derivative of surface area and set it equal to 0. From there, solve for either r or h, depending on which one you plugged in.

anonymous · Answer

Refer to a solution using the Mathematica v9 program.

anonymous · Answer

@robtobey  I completely missed that it was a closed container. Thank you for noting that. 
So, as he stated, 
     $$S = 2\pi rh + 2\pi r^2$$  
The rest of the problem remains the same.