Modeling with basic exponential functions word problem

- [Instructor] 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, 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. Just to remind ourselves what an exponential function would look like, this tells us that our 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 telling us 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 one 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 two to F of one should be equal to the ratio of F of three to F of two, which should be the same as the ratio of F of four to F of three. Or in general terms the ratio of F of t plus one 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 two to F of one? 450 divided by 300, well that's 1.5. That's 1.5. 675 divided by 450, that's 1.5. 1,012.5 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 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 one, well that's just going to be a times r to the t plus one 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 one 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 whether 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 one is. When t is equal to one, 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, but let's do that. F of one 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 one 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. It's 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 zero. Or another way of thinking about it, we need to figure out her fine for t equals zero. So we need to figure out F of zero. So what's F of zero? It's 200 times 1.5 to the zero power. 1.5 to the zero power is one, 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 675 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 one to t equals zero 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 values, or successive months, every time we are multiplying by the common ratio. Every time we are multiplying, we are multiplying by the common ratio.