While I'm sympathetic to the goal of finding so-called interesting problems, Rex's example is hardly an improvement on the textbook variety! <grin>
If the goal is to have kids realize that maxima (often) occur at relative extrema, I think the textbook problems are perfectly adequate. If the goal is to make them combine that fact with other given information, (max-min + related rates), then there's unexplored territory here, but it's easy to frustrate kids with problems where there's a trick involved. If the idea is to give them a problem where the objective function or the constraint needs to be constructed with some creativity, it has to be realized that, yes, these are interesting, but they're not learning "calculus" per se, they're learning creativity. What do we really want? I have to admit that, being as clever as I am, I'm heavily biased in favor of clever problems. But i try to resist!
>>> "Rex Boggs" <firstname.lastname@example.org> 10/25/01 08:58AM >>> We are currently doing max-min problems using derivatives, and I am noticing how unrealistic, trivial or pointless many of the traditional questions are, e.g. inscribing shape A into shape B so as to maximise its area or volume, designing a running track so the area it encloses is a maximum, building chicken pens against barns and rivers, etc.
Are there some max-min problems out there that are more realistic and more interesting than these?
As a starter, I will offer one (based on a real cereal box):
My breakfast cereal comes in a box which measures 160 mm by 75 mm by 245 mm, and it contains 525 grams of cereal.
a. Re-design the box so that it is still rectangular, contains the same amount of cereal, but uses less cardboard in its construction.