Modeling with basic exponential functions word problem

Sarah Swift got a speeding ticket on her way from 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 should be 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 infer 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 the the the reason why are right over here is called is called the common ratio is it's the ratio that if you look at any to say if you were to increment T by one the ratio of that to F of T that ratio is should be consistent for any T so let me give you an example here the ratio 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 should 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 one point five that's one point five six hundred seventy five divided by 450 that's one point five one thousand twelve divided by point 5/6 7 675 that's one point five so the common ratio in all of these situations is one point five so the common ratio over here is one point five and another way and just two and just to make it clear why this our 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 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 1 minus T which is just going to be equal to R to the first power which is just equal to R so this term this this 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 but 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 point 5 and we get a is equal to 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 and euros for Sarah 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 one point it's 200 times one point five to the zero power one point five to the zero power is one so that's just going to be equal to equal to 200 euro now another way of thinking about it is well 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 are dividing by the common ratio so then to go to from to go from T equals 1 at T equals 0 you would divide by the common ratio again and you would get 2 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