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# Constructing exponential models

CCSS.Math:

## Video transcript

dereck 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 want to have a go at this alright 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 0 when it's been 0 weeks since Derek initially sent the chain letter how many people have gotten it when they tell us he initially 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 have received 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 if we factor out a 40 40 times 1 plus 9/10 or you could say this is equal to 40 times whoops 40 times 1.9 or another way of thinking about it after three weeks we've grown 90 percent 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 percent 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 is six weeks well how many people will have received email was going to be this number 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 one point nine times one point nine we're going to grow by another nine-tenths growing by nine tenths is the same thing as multiplying by one and nine-tenths the one is what you already are and then you're growing by another nine-tenths so this is the same thing as 40 times 1.9 squared you go another three weeks nine weeks we're going to grow another 90 percent so you're going to multiply by you're going to 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 it's an exponential function we have our initial value and every three weeks we're multiplying by one point nine so one point nine would be our common ratio so we could say that P of T is equal to our initial value for T times our common ratio one point nine and we multiplied by one point nine 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 the number of three week periods that have gone by and there you have it and notice T equals zero one point nine to the zeroth power is one so forty times one T equals three that's going to be one point nine to the first power three over three and so we're going to grow by ninety percent and so on and so forth so feeling pretty good about this