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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.