# Suggest some optimization problems for me, please.

1. Oct 28, 2009

### SpEuler

Hi, this is my first post and most certainly not my last. I'm a young Mechanical Engineering major and I love math and physics, but on with my topic...

I'm in Calc I and we've been assigned an extra-credit group project where to do present either a related rate or an optimization problem. Everyone in my group is some sort of engineering major and we're all fairly competent with math and physics. We want to do something different that's not in our book; if we have to maximize the volume of one more box, we'll hang ourselves. I was hoping to get some suggestions for some fun, but challenging optimization problems (or related rates if you can think of very cool ones). Problems that involve physics are preferred.

Thanks.

Oh and don't take the fun out of it and post answers, but relevant formulas are welcome. =)

2. Oct 28, 2009

### Tac-Tics

There's all sorts of things you can extremize. I can't say how hard any of them are.

Given two curves (surfaces) what two points are the closest together (farthest apart)?

Given a model of supply and demand, what is the most profitable price to sell product?

Given a model of your education's worth to the cost of school over time, at what point does staying at your university hit the point of diminishing returns?

If you're doing physics, legrangian mechanics is all about turning motion into an optimization problem, extremizing the action of a system.

Come up with a model of an eco system. How many frogs can you add to the environment in order to keep the number of flies to a minimum without critically disrupting any other wildlife?

You're a dictator of a country (CEO of a company). How much can you tax (treat) your peasants (employees) before they revolt (go on strike)?

3. Oct 28, 2009

### Hurkyl

Staff Emeritus
Let $f(x) = 2^x - x^2$.
Minimize $f'(x)$ over the interval $[4, +\infty)$.
Does that tell you anything interesting?

(The optimization isn't particularly challenging, but I think the application is pretty neat)

4. Oct 28, 2009

### l'Hôpital

Find the biggest cone you can fit in a sphere of radius r.