Constructing exponential models according to rate of change
- [Voiceover] 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 9/10 of its size every three 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 wanna have a go at this. All right, now the way I'd like to think about this, let's just gonna 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? Well, they tell us. He initially, Derek initially sent the chain letter to 40 friends. So t equals zero. P of t or P of zero is 40. Now, what's an interesting time period? It says that the email, the number of people who receive the email gains 9/10 or increases by 9/10 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 9/10 of its size. So it's going to be 40 plus 9/10 times 40 which is going to be equal to, what? Well, that's equal to 40. We factor out our 40. 40 times one plus 9/10 or you could say this is equal to 40 times, whoops, 40 times 1.9. Another way of thinking about it, after three weeks, we've grown 90%. That's another way of saying that the number of people who receive the email gains 9/10 of its size. You could say the group of people who receive the email grows 90% every three weeks. And so if we go another three weeks, so plus another three weeks, I could say, well, let me just write this, this is six weeks. Well, how many people would have received the email? It was going to be this number. It's going to be grown another 90%. So we're gonna multiply it times 1.9 again. So it's going to be 40 times 1.9 times 1.9. We're gonna grow by another 9/10. Growing by 9/10 is the same thing as multiplying by one and 9/10. The one is what you already are. Then you're growing by another 9/10. So this is the same thing as 40 times 1.9 squared. We go another three weeks, nine weeks. We're gonna grow another 90%. So you're gonna multiply it by, you're gonna take this number and multiply by 1.9 again which is going to be 1.9 to the third power. And so what's going on over here? Well, we can see some exponential function. We have our initial value. And every three 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 three week periods have passed by? Well, we will take t and divide it by three. T divided by three is a number of three week periods that have gone by. And there you have it. And notice, t equals zero, 1.9 to the zeroth power is one so 40 times one. T equals three, that's gonna be 1.9 to the first power, three over three. And so we're gonna grow by 90% and so on and so forth. So feeling pretty good about this.