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- Let's do a couple of word problems dealing with
- exponential growth and decay.
- So this first problem, suppose a radioactive substance decays
- at a rate of 3.5% per hour.
- What percent of the substance is left after 6 hours?
- So let's make a little table here, to just imagine
- what's going on.
- And then we'll try to come up with a formula for, in
- general, how much is left after n hours.
- So let's say hours that have passed by,
- and percentage left.
- So after 0 hours, what percent is left?
- Well, it hasn't decayed yet, so we have 100% left.
- After 1 hour, what's happened?
- It decays at a rate of 3.5% per hour.
- So 3.5% is gone.
- Or another way to think about it is 0.965.
- Remember, if you take 1 minus 3.5%, or if you take 100%
- minus 3.5%-- this is how much we're losing every hour-- that
- equals 96.5%.
- So each hour we're going to have 96.5% of
- the previous hour.
- So in hour 1, we're going to have 96.5% of hour 0, or 0.965
- times 100, times hour 0.
- Now, what happens in hour 2?
- Well, we're going to have 96.5% of the previous hour.
- We will have lost 3.5%, which means that we have 96.5% of
- the previous hour.
- So it'll be 0.965 times this, times 0.965 times 100.
- I think you see where this is going, in general.
- So in the first hour, we have 0.965 to the first
- power, times 100.
- In the zeroth hour, we have 0.965 to the zeroth power.
- We don't see it, but there's a 1 there, times 100.
- In the second hour, 0.965 to the second power, times 100.
- So in general, in the nth hour-- let me do this in a
- nice bold color-- in the nth hour, we're going to have
- 0.965 to the nth power, times 100 left of
- our radioactive substance.
- And oftentimes you'll see it written this way.
- You have your initial amount times your common ratio, 0.965
- to the nth power.
- This is how much you're going to have left after n hours.
- Well, now we can answer the question.
- After 6 hours how much are we going to have left?
- Well, we're going to have 100 times 0.965 to
- the sixth power left.
- And we could use a calculator to figure out what that is.
- Let's use our trusty calculator.
- So we have 100 times 0.965 to the sixth power, which is
- equal to 80.75.
- This is all in percentages.
- So it's 80.75% of our original substance.
- Let's do another one of these.
- So we have, Nadia owns a chain of fast food restaurants that
- operated 200 stores in 1999.
- If the rate of increase is-- oh actually, there's a typo
- here, it should be 8%-- the rate of increase is 8%
- annually, how many stores does the
- restaurant operate in 2007?
- So let's think about the same thing.
- So let's say years after 1999.
- And let's talk about how many stores Nadia is operating, her
- fast food chain.
- So 1999 itself is 0 years after 1999.
- And she is operating 200 stores.
- Then in 2000, which is 1 year after 1999, how many is she
- going to be operating?
- Well, she grows at the rate of 8% annually.
- So she'll be operating all the stores that she had before
- plus 8% of the store she had before.
- So 1.08 times the number of stores she had before.
- And you're going to see, the common ratio here is 1.08.
- If you're growing by 8%, that's equivalent to
- multiplying by 1.08.
- Let me make that clear.
- 200 plus 0.08, times 200.
- Well, this is just 1 times 200 plus 0.08, times 200.
- That's 1.08 times 200.
- Then in 2001, what's going on?
- This is now 2 years after 1999, and you're going to grow
- 8% from this number.
- You're going to multiply 1.08 times that number,
- times 1.08 times 200.
- I think you get the general gist. If, after n years after
- 1999, it's going to be 1.08-- let me write it this way.
- It's going to be 200 times 1.08 to the nth power.
- After 2 years, 1.08 squared.
- 1 year, 1.08 to the first power.
- 0 years, this is the same thing as a 1 times 200, which
- is 1.08 to the zeroth power.
- So they're asking us, how many stores does the restaurant
- operate in 2007?
- Well, 2007 is 8 years after 1999.
- So here n is equal to 8.
- So let us substitute n is equal to 8.
- The answer to our question will be 200 times 1.08 to the
- eighth power.
- Let's get our calculator out and calculate it.
- So we want to figure out 200 times 1.08
- to the eighth power.
- She's going to be operating 370 restaurants, and she'll be
- in the process of opening a few more.
- So if we round it down, she's going to be operating 370
- restaurants.
- So 8% growth might not look like something that's so fast
- or that exciting.
- But in under a decade, in only 8 years, she would have gotten
- her restaurant chain from 200 to 370 restaurants.
- So over 8 years, you see that the compounding growth by 8%
- actually ends up being quite dramatic.