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Quadratic and exponential word problems — Basic example

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- [Instructor] A cable company with a reputation for poor customer service is losing subscribers 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 (chuckles). 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 zero, when t equals zero, this is zero years after the start of 2014. So this would be the number of subscribers they had at the start of 2014, which is two million. So S of zero is going to be two. Remember, S is in terms of millions, they tell us that. So S of zero is two. What is S of one going to be, when t equals one? Well, one year has gone by, so they're going to lose 3% of their subscribers, and losing 3% is the same thing as retaining 97%. So it's going to be two times 0.97. Now what happens at t equals two, after two years? Well, they started with two million. In one year, they were able to, only to retain 97%. And then another year goes by. They're only gonna retain 97% of what they had or after, what they had the year before, so another 97%. So I see, I think you see the trend. You're going to multiply by 97% as many times or as t times I guess is another way to think about it. If you say S of three, you started with two million subscribers, after one year you retain 97% of them, after another year you're gonna retain 97% of this, and after another year, at t equals three, you're gonna retain 97% of that, 97% of that. So in general, S of t, it's going to be what you started with times 0.97 to the t-th power. However many years have gone by, you take your, I guess you'd say your retention rate to that power. And of course, 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've done it is you could've 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 1.03 times 1.03, it's going to get larger than one. You're gonna have more than two million subscribers, so as t increases, so you could rule that one out. In this one, every year, 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 linear, this is kind of a, well, it's, I mean, they're saying you're multiplying by t. And, you know, one way to realize that this is gonna break down very fast is t equals zero, this is gonna 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 one's gonna increase as t increases, while we need to have a decreasing number of subscribers, so you could rule that one out as well.