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# Modeling with basic exponential functions word problem

Sal solves a word problem where he models the growth of a speeding ticket fine over time as an exponential function, and then interprets this model. Created by Sal Khan.

Video transcript

Sarah Swift got a speeding
ticket on her way home from work. If she pays her fine now,
there will be no added penalty. If she delays her
payment then a penalty will be assessed for
the number of months t that she delays paying her fine. Her total fine F in euros is
indicated in the table below. These numbers represent
an exponential function. So they give us the
number of months that the payment is delayed,
and then the amount of fine. And this is
essentially data points from an exponential function. And just to remind ourselves
what an exponential function would look like,
this tells us that are fine as our function
of the months delayed is going to be equal to some
number times some common ratio to the t power. This exponential
function is essentially telling us that our
function is going to have this form
right over here. So let's see if we can
answer their questions. So the first question is,
what is the common ratio of consecutive values of F? So the reason why
r right over here is called the common ratio is
it's the ratio that if you look at any two-- say if
you were to increment t by 1, the ratio of
that to F of t-- that ratio should be
consistent for any t. So let me give you
an example here. The ratio of F of
2 to F of 1 should be equal to the ratio
of F of 3 to f of 2, which would be the same as
the ratio of F of 4 to F of 3. Or in general terms,
the ratio of F of t plus 1 to the
ratio of F of t should be equal to
all of these things. That would be the common ratio. So let's see what that is. If we just look at the form. If we just look at
this right over here. So what's the ratio
of F of 2 to F of 1? 450 divided by 300? Well that's 1.5. 675 divided by 450? That's 1.5. 1012.50 divided by 675? That's 1.5. So the common ratio in all
of these situations is 1.5. So the common ratio
over here is 1.5. And another way-- and
just to make it clear why this r right over here
is called the common ratio-- is let's just do
this general form. So f of t plus 1? Well that's just going to be a
times r to the t plus 1 power. And F of t is a times
r to the t power. So what is this going to be? This is going to
be-- let's see-- this is going to be r to the t
plus 1 minus t, which is just going to be equal to r to the
first power, which is just equal to r. So this variable r is going to
be equal to this common ratio. So when we figured out
that the common ratio is 1.5, that tells us that
our function is going to be of the form F of t is
equal to a times-- instead of writing an r
there, we now know that r is 1.5 to the t power. 1.5 to the t power. Write a formula
for this function. Well we've almost
done that, but we haven't figured out what a is. And to figure out what a is,
we could just substitute-- we know what F of 1 is. When t is equal to
1, F is equal to 300. And so we should be able
to use that information to solve for a. We could have used any of these
data points to solve for a. So let's do that. F of 1 is equal to a times
1.5 to the first power, or a times 1.5. And that is going to be equal
to-- they tell us that F of 1 is equal to 300. And so another way
of writing this is we could say 1.5 times
a is equal to 300. Divide both sides by 1.5. And we get a is equal to 200. And so our function, our
formula for our function is-- let me write
it in black so we can see it-- is going to be
200-- that's our a-- times 1.5 to the t power. Now another way--
well actually let's just think about
the next question. What is the fine in euros
for Sarah's speeding ticket if she pays it on time? So paying it on time, that
implies that t is equal to 0. Or another way of
thinking about it, we need to figure out
her fine for t equals 0. So we need to figure out F of 0. So what's F of 0? It's 200 times 1.5
to the 0 power. 1.5 to the 0 power
is 1, so that's just going to be equal to 200 euro. Now another way of thinking
about it is, well look, let's look at the common ratio. To go from 6.5 to 450,
you're essentially dividing by the common ratio. To go from 450 to 300, you're
dividing by the common ratio. So then to go from
t equals 1 to t equals 0, you would divide
by the common ratio again. And you would get to 200. Or another way of
thinking about it is to go to successive
months every time we are multiplying
by the common ratio. Every time we are multiplying
by the common ratio.