* “The BBC reports that the rate of deforestation in the Amazon has halved”:http://news.bbc.co.uk/1/hi/world/americas/4189792.stm
* Last year the rainforest lost 9,000 Km^2^, compared with roughly 18,000 Km^2^ in previous years
Halving the _rate_ of change does not stop the forest shrinking, but the shrinking will slow down. Many environmental organisations point out that these figures can’t show a permanent trend yet. The Brazillian government hopes that the fall in destruction of the rainforest is in response to a tightening of the regulations.
Suppose we say the annual rate continues to fall by a _factor_ of two each year. Starting at 18000, we get a series like 9000 and 4500, 2250, 1125, 562.5, 281.25 and so on. Each successive year is adding in a smaller contribution to the total loss. After 10 years, you are looking at a loss of 18 Km^2^ or roughly one thousandth of the loss in 2003 (2^10^ = 1024). This series is called a “geometric progression”:http://en.wikipedia.org/wiki/Geometric_progression that happens to have a fractional ‘growth rate’ – like compound interest in reverse.
Plotting the series against year in Excel, you get something like
If we do a running subtotal of the loss assuming that the loss is halved each year – rather like a cumulative frequency curve in statistics, the result looks something like this…
As you can see, the total loss will approach around 36 000 Km^2^ and each subsequent year will add a smaller and smaller amount, so that the total will never _quite_ reach twice the starting figure. The sequence _tends to a limit_ of twice the starting rate.
h3. Challenges
* can you devise a spreadsheet that will calculate the deforestation for each year and the cumulative total?
* If the deforestation actually ‘grows’ by a factor of 0.75 instead of 0.5 each year, what will the new limit be? Use your spreadsheet to find out. You may need to change the formulas
* Can you use the “formula from the Wikipedia page”:http://en.wikipedia.org/wiki/Geometric_progression for the sum of an infinite geometric progression to work out what the maximum loss to the rainforest will be based on a growth rate of 0.5 per year? hint – write the sequence like 18000 × ( 1 + 0.5 + 0.25 + 0.125…) and identify X in the wikipedia formula…
We have built a simple _mathematical model_ of deforestation of the Amazon rainforest. Alas I suspect the assumption of halving each year is going to be too optimistic. We can hope.
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