**An Example of Extrapolation:**

**Overpopulation on Mars**

Suppose we started a

self-sustaining colony

100 colonists

What is your estimated

rate of increase per year?

What is the total population

capacity of Mars?

Earth surface area = 510,072,000 km^2

Earth land area = 148,940,000 km^2

Mars surface area = 144,798,500 km^2

(0.284 of Earth)

How long do you think it will take

for Mars to overpopulate?

We can check this using a spreadsheet!

Just have the rows represent successive years

Each year has x% more people than the previous

See how many years go by until overpopulation!

**Making and Discuss Predictions with**

*Trajectories*Method: Trajectories of change

. . . in the short term,

change appears

**linear**

Example:

Last year you had 1 or 2 compact fluorescent bulbs

This year you will "probably have 1-2 more"

In the longer term,

change may looks

**exponential**

Lightbulb example:

. . . you start with 1-2, but after a couple of years you've got a bunch

. . . change accelerates, in this case

. . . if you look at an exponential curve with a microscope, what does it look like?

. . . "Exponential": complicated word, tricky math, simple concept

. . . . . . goes up faster and faster

. . . . . . has a doubling time

**Exponential curves explained**

. . . Suppose something doubles every 3 years

. . . Popular example: computer CPU complexity doubles every 2 years

. . . new value after

*t*years is original value

*v*times 2^(t/3)

. . .

*f*(

*t*)=

*t*o * 2^(

*t*/3)

. . . . . . where does the "doubles" appear?

. . . . . . where does the "every 3 years appear?

. . . . . . so it works for

any factor of increase and

any time constant

**Longer term, things "level off": the S-curve**

Also called "logistic curve"

Sort of "linear" early on

Then looks "exponential"

Then levels off

Justified by many, many diverse phenomena modelable as:

Malthusian scenarios

Constructal Theory scenarios

A. Bejan and S. Lorente, The constructal law origin of the logistics S curve, Journal of Applied Physics, vol. 110 (2011), 024901, www.constructal.org/en/art/S-curve.pdf.

**Do you think an even longer-term view will look like a plateau curve?**

Think about pencils, compact fluorescents, and college, etc., etc.

What do you think of these curves?

(Source: http://nextbigfuture.com/2010/08/white-led-lights-with-135-lumens-per.html, 9/5/10)

There are other view of trajectories...

Gartner Hype Cycle

(source: http://en.wikipedia.org/wiki/File:Gartner_Hype_Cycle.svg)

How might this apply to some of our topics?

There is also the Technology Adoption Life Cycle

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