MathBench > Population Dynamics

Exponential Growth and Decay

Growth of the computer industry

When the home computer first came on the market in the late 1970s there were many people who pooh-poohed the idea. Why would anyone possibly want a computer in their home? Boy, were they wrong (and possibly a little embarrassed)! By the mid 1990s 1/3 of US homes had a computer.

Early computerIn 1997, 37% of homes had a computer. If computers expanded by 12% annually, how many homes had them in 2000?

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I think I have the answer: 37×1.123= 52%

The Internet also had its naysayers, old netscape diskbut its usage has grown even faster (32% per year). Starting at 18% of homes in 1997, what would you expect in 2000?

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I think I have the answer: 18×1.323= 41%

If these rates continued, in what year would the percent of homes with Internet be approximately equal to the number of homes with a computer?

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I think I have the answer: Sometime between 2001 and 2002

By the way, if you're a bit disturbed by the last calculation, you have a right to be. great aunt jeaneneWhy? Because the percentage of homes with Internet should not really be able to exceed the percentage of homes with a computer. So if the above model is correct, then

These three unlikely conditions are shown in the graph below. So is exponential growth "wrong"? Well, the model itself isn't wrong, but we may have applied it a bit too eagerly. Assuming the growth rates stay constant over long periods of time doesn't really make sense.

If, instead, we assume that growth rates gradually decline (both for computer ownership and for Internet access) we get more reasonable results -- which you can see by rolling your mouse over the graph. This is still exponential growth, but the growth rate is no longer constant -- that's why the lines don't curve up.

graphs showing the model

The moral of the story? Math is fun and all, but don't throw out your common sense.