[This is a draft placeholder for book exercises about the phosphorus pollution model]

Visit the Geogebra simulator for a model of phosphorus pollution.

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# Mathematics for Sustainability

## Phosphorus Pollution Model

## Budyko Climate Model

## The Betz limit

## Sample question and response

## Page 336 Sea Ice Data

All NSIDC data is available for download here.

measuring, flowing, networking, changing, risking, deciding

[This is a draft placeholder for book exercises about the phosphorus pollution model]

Visit the Geogebra simulator for a model of phosphorus pollution.

[This is a draft placeholder for book exercises about the Budyko climate model]

Visit the Geogebra simulator for the Budyko climate model, incorporating the ice-albedo feedback.

There is an interesting discussion to be had about the possible efficiency of wind turbines which presents opportunities for the instructor to look at several different kinds of computations. The discussion ends up with an optimization problem which could be approached by the classical methods of calculus – if the students have that available to them – or could be approximated simply by drawing a graph of one critical function. Continue reading “The Betz limit”

A reader asked “Why did the chicken cross the road?”

It is well known that this is a hard question to analyze rigorously. However, by starting with the 1-dimensional chicken diffusion equation

\[ \frac{\partial^2\chi}{\partial x^2} + \frac{1}{4\sqrt{\pi}}\frac{\partial \chi}{\partial t} \approx 0, \]

where \(\chi\) is the local density of chickens, we may arrive at the answer, “Because it wanted to get to the other side (to a first order approximation).”

A couple of exercises for chapter 5 refer to a dataset of Arctic sea ice extent, and we also plan to add an online case study on *regression to the mean* where this will be one of the examples. The data comes from the National Snow and Ice Data Center *Arctic Sea Ice News and Analysis*. The dataset is more completely summarized in the graphic below.