# Design a Tin Can (harder pre-cal but easy calc)

Design a tin can so that the total area including a top and bottom is minimized for a required volume. The can has radius *r* and height *h*.

The reason for changing variables was to choose a nice *a* which will make *u* unitless and make the problem generic since the numerical value of *V* has nothing to do with the essence of the problem. Also, we can get rid of the π so that the function to minimize is easy to study. The value of *a* **landen** picked is not as arbitrary as it looks at first and there was some experimentation with the substitution using Maxima (*http://maxima.sourceforge.net*) to find a good one.

Now all we need to do is minimize:

A little playing around with a calculator suggest that the minimum value is *f*(*u*) = 3 and that this happens when *u* = 1. So we rewrite f(u) to emphasize 3 and 1 and we hit the jackpot.

In this form it is obvious that the minimum value of *f*(*u*) = 3 and this happens when *u* = 1.