Interesting, unusual max/min problems?

Question

So I've got to that stage of my elementary mathematics subject for engineers when we talk about differentiation and solution of max/min problems. And I'd like to entertain and engage the students with some interesting problems. Pretty much every book and website talks about maximizing rectangular areas of land with fences of a given length, or maximizing the volume of a box with square cross section (and given surface area) etc. No student is ever excited by these. Slightly more interesting is the problem of maximizing the length of a pipe (considered as having zero cross section) which can be manoeuvred around a right-angled hallway. There must be more interesting problems, or at least, more interesting ways to dress up these problems.

But I don't know of any, and would be delighted to know of some!

Answer 1

The examples you gave my themselves are elementary but good examples already. However, if you want to expand even more, you have many options. I'll name the ones I can think of at the moment.

1. Economics has a lot of great maximization problems at various levels, especially microeconomics.
2. Physics, chemistry, and biology use optimization problems a lot. An interesting outside-look of optimization (not your standard AP calculus optimization) are out-of-the-box things like these.
3. If you want you want more math-related optimization, multi-variable optimization is not very difficult to introduce and it is at least slightly more interesting. The transition from single-variable calculus to multi-variable calculus can be made very smooth such that the students do not even realize that they are doing higher math.
4. Come up with your own! It's not very difficult to come up with interesting scenarios on your own. For example:

   • A squirrel is walking along number line. Its position is given by the function $ x = \sin \left( 2014t \right) $, and $t$ is the elapsed time. What is the first time $t>0$ at which the squirrel stops?

Of course, the squirrel example is very basic but I'm just setting an example which you can build upon.

Answer 2

This is another rectangular problem, but I like it because the student can take one of at least 3 approaches.

A spider in on the floor in the north-west corner of a room. He would like to crawl to the south-east corner on the ceiling. What is the path that minimizes the distance that he has to crawl?

Possible approaches

1. When the spider reaches the edge of the ceiling and a wall he will change direction slightly. The angular difference in his directions crawled is a function of the distance between a corner and where he crosses the edge. Minimize that angle.
2. When the spider reaches the edge of the ceiling and a wall he will change direction slightly. The distance crawled is a function of the distance between a corner and where he crosses the edge. Maximize that distance.
3. knock down the walls and draw a straight line.

I like to think of myself as being smarter than the average bear, but I didn't choose strategy 3 :(

Answer 3

Two books; I guess yesterday, somebody asked again about the question of Regiomantus, Finding the widest angle to shoot a soccer ball from the sideline using optimization!!

I knew of this from some book i had 40 years ago, but it is in two books that can be purchased (or borrowed) , Heinrich Dorrie (translated) 100 Great Problems of Elementary Mathematics with a cozen problems in the final chapter, called Extremes. Andre recommended Ivan Niven, Maxima and Minima without Calculus; I am having trouble seeing whether that is still in print...