Suggest some optimization problems for me, please.

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Discussion Overview

The discussion revolves around suggestions for optimization problems suitable for a group project in a calculus course, particularly those that incorporate elements of physics and are distinct from standard textbook examples. Participants seek creative and challenging problems that engage their mathematical and engineering skills.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant expresses a desire for unique optimization problems, specifically avoiding common examples like maximizing the volume of a box.
  • Another participant suggests various optimization scenarios, including finding the closest or farthest points between two curves, determining the most profitable price in a supply and demand model, and analyzing the diminishing returns of education costs over time.
  • A suggestion is made to model an ecosystem to find the optimal number of frogs that can be added without disrupting other wildlife.
  • One participant proposes a mathematical function, f(x) = 2^x - x^2, and asks about minimizing its derivative over a specified interval, noting the application may be interesting despite the optimization being straightforward.
  • A challenge is posed to find the largest cone that can fit inside a sphere of a given radius.

Areas of Agreement / Disagreement

Participants generally agree on the need for creative optimization problems, but there is no consensus on specific problems or their difficulty levels. Multiple competing views on potential problems remain, and the discussion is open-ended.

Contextual Notes

Some suggestions may depend on specific interpretations of the problems or the mathematical models used, and the difficulty of the proposed problems is subjective and not universally defined.

Who May Find This Useful

Students in calculus or engineering courses looking for innovative optimization problems, educators seeking examples for teaching purposes, and individuals interested in the application of mathematics to real-world scenarios.

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

(The optimization isn't particularly challenging, but I think the application is pretty neat)
 
Find the biggest cone you can fit in a sphere of radius r.
 

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