Constructing exponential models | Mathematics II | High School Math | Khan Academy

Derek sent a chain letter to his friends, asking them to forward the letter to more friends. The group of people who receive the email gains 910 of its size every 3 weeks and can be modeled by a function P, which depends on the amount of time T in weeks. Derek initially sent the chain letter to 40 friends.

Write a function that models the group of people who receive the email T weeks since Derek initially sent the chain letter. So pause the video if you want to have a go at this.

All right, now the way I like to think about these, let's just create a table with values for T and our function P, which is a function of T, for some values that we can just pull out of the description here.

So when T is zero, when it's been zero weeks since Derek initially sent the chain letter, how many people have gotten it? What they tell us is that Derek initially sent the chain letter to 40 friends. So T equals 0, P of T, or P of 0, is 40.

Now, what's an interesting time period here? It says that the email, the number of people who've received the email, gains 910 or increases by 910 every three weeks. Every three weeks, so after three weeks, so three weeks have gone by.

So, I'm just adding three to T. What is P of T going to be? Well, they tell us it's going to gain 910 of its size. So it's going to be 40 plus 910 times 40, which is going to be equal to what?

Well, that's equal to 40. If we factor out 40, we get 40 * (1 + 910), or you could say this is equal to 40 times... whoops, 40 times 1.9, or another way of thinking about it, after 3 weeks, we've grown 90%.

That's another way of saying that the number of people who receive the email gains 910 of its size. You could say the group of people who receive the email grows 90% every 3 weeks.

And so if we go another 3 weeks, so plus another three weeks, I could say, well, let me just write this as six weeks. Well, how many people will have received the email?

Well, it's going to be this number, and it's going to be grown another 90%. So we're going to multiply it times 1.9 again. So it's going to be 40 times 1.9 times 1.9.

We're going to grow by another 910. Growing by 910 is the same thing as multiplying by one and 910. The one is what you already are, and then you're growing by another 910.

So this is the same thing as 40 * (1.9 squared). You go another 3 weeks, nine weeks, where you're going to grow another 90%. So you're going to take this number and multiply by 1.9 again, which is going to be 1.9 to the 3rd power.

And so what's going on over here? Well, we can see it's an exponential function. We have our initial value, and every 3 weeks, we're multiplying by 1.9.

So 1.9 would be our common ratio. So we could say that P of T is equal to our initial value, 40, times our common ratio, 1.9.

And we multiply by 1.9 every three weeks. So we could just say how many 3-week periods have passed by. Well, we would take T and divide it by three.

T divided by three is the number of 3-week periods that have gone by. And there you have it. Notice T equals 0, 1.9 to the 0th power is 1.

So 40 * 1, T = 3, that's going to be 1.9 to the 1st power. 3 over 3, and so we're going to grow by 90% and so on and so forth. So feeling pretty good about this.