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a cable company with a reputation for poor customer service is losing subscribers at an at a rate of approximately 3% per year the company had two million subscribers at the start of 2014 assume that the company continues to lose subscribers at the same rate and that there are no new subscribers it's truly a bad situation for them which of the following functions s models the number of subscribers in millions remaining t years after the start of 2014 so let's just think about this a little bit so s of 0 when T equals 0 this is 0 years after the start of 2014 so this would be the number of subscribers they had at the start of 2014 which is 2 million so s of 0 is going to be 2 remember s is in terms of millions they tell us that so s of 0 is 2 what is s of 1 going to be when t equals 1 well one year has gone by so they're going to lose 3 percent of their subscribers and losing 3 percent is the same thing as retaining 97% so it's going to be 2 times zero point nine seven now what happens at t equals 2 after 2 years well they started with 2 million in one year they were able to only to retain 97% and then another year goes by they were only gonna retain 97 percent of what they had or after what they had the year before so another 97 percent so I see I think you see the trend you're going to you're going to multiply by 97 percent as many times or as T times I guess there's another way to think about it if you say s of 3 you started with 2 million subscribers after one year you retain 97 percent of them after another you're gonna retain 97 percent of this and after another year that equal story you're gonna have retained 97 percent of that 97 percent of that so in general S of T it's going to be what you started with times 0.9 7 to the T power however many years have gone by you take your I could say retention rate to that power and you then multiply that times your initial starting subscribers and that's how much you're gonna be left with and let's see which of these choices have that that is this choice right over here now another way you could have do and dot done it is you could have tried to rule out some choices here this one actually has the subscribers growing if you multiply by 1.03 to the t 1.03 times 1.0 3 times 1.3 it's going to get larger than 1 you're gonna have more than 2 million subscribers so as T increases so you could rule that one out in this one every year you're you're only retaining 70% of your subscribers you're losing 30% not 3% so that one's even worse than this already bad situation and then this one this is a a linear this is this is kind of a well it's I mean they're saying you're multiplying by T and you know one one way to realize that this is going to break down very fast is T equals zero this is going to give us zero but at T equals zero you don't have zero subscribers at T equals zero you have two million subscribers the other way to think about it is this was going to increase as T increases while we need to have a decreasing number of subscribers so you could rule that one out as well