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August 22nd, 2018, 09:44 AM   #1
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Need help with a maths project!

I have a maths project in my school, where I have to come up with a topic and a problem and investigate it using maths. The level of maths that this project should be at is high school and pre-college.

I was thinking of formulating some mathematical model on an aspect of real life, but I'm not sure what to model and how to make it of a complex level.

Modelling populations is a topic I thought of, but I'm not sure how to add on to it apart from just coming up with a function.

Any suggestions/help?

Thank you very much.

Last edited by skipjack; August 22nd, 2018 at 11:50 AM.
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August 22nd, 2018, 09:50 AM   #2
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I don't think differential equations are pre-college so that's going to limit what you can model.

One thing that comes to mind that is easy enough to collect data on is the time it takes a known volume of water at a known initial temperature to reach a boil.

By controlling the volume of water and the initial temperature carefully you can isolate out the time as a function of volume.

You'll want to use the same pot for all the volumes to keep the amount of water exposed to the air constant as well.

You might try the same experiment with and without a lid to see what that effect might be.
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August 22nd, 2018, 10:05 AM   #3
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One thing that comes to mind that is easy enough to collect data on is the time it takes a known volume of water at a known initial temperature to reach a boil.
NO NO: should be "to become frozen"
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August 22nd, 2018, 10:37 AM   #4
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Quote:
Originally Posted by davidprince View Post
I have a maths project in my school, where I have to come up with a topic and a problem and investigate it using maths. The level of maths that this project should be at is high school and pre-college.

I was thinking of formulating some mathematical model on an aspect of real life, but I'm not sure what to model and how to make it of a complex level.

Modelling populations is a topic I thought of, but I'm not sure how to add on to it apart from just coming up with a function.

Any suggestions/help?

Thank you very much.
This is a starting point for a model on population growth that uses nothing but simple algebra and a recurrence relation, but introduces the topic of chaos theory.

https://en.wikipedia.org/wiki/Logistic_map

That wiki article quickly gets technical, but watch the little movies there to get a flavor of what happens as r changes. There is a good popularization of the field in a book called Chaos, written by Gleick, that should be available through a library.

Last edited by skipjack; August 22nd, 2018 at 11:51 AM.
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August 22nd, 2018, 10:58 AM   #5
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One thing that comes to mind that is easy enough to collect data on is the time it takes a known volume of water at a known initial temperature to reach a boil.
Depends on where that boil is.
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August 22nd, 2018, 11:02 AM   #6
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Depends on where that boil is.


How do you make holy water?

You take normal water, and boil the hell out of it.

Last edited by Denis; August 22nd, 2018 at 11:04 AM.
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August 22nd, 2018, 11:30 AM   #7
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Originally Posted by JeffM1 View Post
This is a starting point for a model on population growth that uses nothing but simple algebra and a recurrence relation, but introduces the topic of chaos theory.

https://en.wikipedia.org/wiki/Logistic_map

That wiki article quickly gets technical, but watch the little movies there to get a flavor of what happens as r changes. There is a good popularization of the field in a book called Chaos, written by Gleick, that should be available through a library.
Sounds like something I could go through. Could you explain further how I would apply it to model a population of a country?
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August 22nd, 2018, 01:45 PM   #8
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Depends on where that boil is.
I have to admit I assumed OP if doing this would use the same stove for each trial and maintain a constant altitude.

Varying this could be for the next project!
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August 22nd, 2018, 08:28 PM   #9
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Sounds like something I could go through. Could you explain further how I would apply it to model a population of a country?
The model is way too simple to use for human populations (though Malthus tried in a half-hearted way to do so). The major problem with trying to apply the model to human populations is that most humans have long-term foresight and that human societies discourage unrestrained reproduction.

In other words, it is a model that probably oversimplifies even animal populations. I have forgotten the simplest model of animal population dynamics that has been strongly verified empirically; if I remember correctly, it turns out to be a catastrophe theory model, not a chaos theory model, involving the interactions of predator and prey populations. So I was not recommending the logistic map as an actual predictive model for any population. It gives clues as to why animal populations may not be stable over time, and the math is easily within the grasp of a high school student. But even animal populations are subject to mutiple influences, and a robust predictive model must take all the major influences into account.

Sadly, the best models of human (and bacterial) population growth appear to be exponential models, which are not very interesting mathematically and are obviously short term models (short term here meaning with respect to humans a time frame measured in centuries). A second year algebra book will give you what you need to know about exponential functions.

EDIT: When I was in college, my room mate did a senior project on growth rates of different strains of bacteria subjected to different dosages of penicillin. They all showed growth rates very close to exponential (at different rates of course) until they reached the edge of the Petri dish. You could easily find materials on growth rates of bacteria that are mathematically tractable. The resulting models are pertinent both to the progression of infection and, with the addition of probability theory, epidemiology.

Last edited by JeffM1; August 22nd, 2018 at 08:43 PM.
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August 23rd, 2018, 06:32 AM   #10
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....Jeff, shouldn't a "sale of condoms factor" be used ....?
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