# Growth of the computer industry

When the home computer first came on the market in the late 1970's 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 1990's 1/3 of US homes had a computer.

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

(To make this problem interactive, turn on javascript!)

- I need a hint: Find the multiplier is 1 + 0.12 = 1.12
- ... another hint ... : This is growth over a three year period, so raise the multiplier to the third power
- ... another hint ... : Starting value is 37

#### I think I have the answer: 37×1.12^{3}= 52%

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

(To make this problem interactive, turn on javascript!)

- I need a hint: What's the multiplier?
- ... another hint ... : how many years?
- ... another hint ... : Where did we start?

#### I think I have the answer: 18×1.32^{3}= 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?

(To make this problem interactive, turn on javascript!)

- I need a hint: Keep increasing the number of timesteps and recalculate
- ... the next calculation... : 37×1.12
^{4}vs 18×1.32^{4} - ... another hint ... : Answer will be approximate

#### 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**. Why? 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

- at the moment when % of homes with computers first equalled % of homes with Internet, the growth of Internet must have slowed to be no more than the growth of computers, and
- after 2002, every home with a computer also had Internet, which we know is not true (like my great aunt Jeanene in Michigan, who uses her Tandy to play mah-jong), and
- in 2006, the number of homes with computer and Internet exceeded 100% for the first time in human history (alert the Onion!)

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.

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

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