The Most Efficient Cylinder
The generic equation for the surface area of a cylinder is
Since multi-variable equations aren't of much use here, h can be rewritten in terms of r as
. This yields an equation of
This equation is then graphed in order to find the point where surface area is the smallest, with X being radius and Y being surface area.
Using the calc-minimum function of the TI-83/84, the vertex is found to be approximately
. Thus, the radius is plugged back into the equation for height to find that the corresponding height is
Therefore, the ratio of the radius to the height is