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Video transcript

- [Instructor] We are told a phone sells for $600 and loses 25% of its value per year. Write a function that gives the phone's value, V of t, so value as a function of time, t years after it is sold. So pause this video, and have a go at that before we work through it together. All right, so let's just think about it a little bit. And I could even set up a table to think about what is going on. So this is t, and this is the value of our phone as a function of t. So it sells for $600. At time t equals zero, what is V of zero? Well it's going to be equal to $600. That's what it sells for at time t equals zero. Now at t equals one, what's going to happen? Well it says that the phone loses 25% of its value per year. Another way to rewrite that it loses 25% of its value per year is that it retains, it retains, 100% minus 25% of its value per year, or it retains 75% of its value per year, per year. So how much is it going to be worth after one year? Well it's going to be worth $600, $600 times 75%. Now what about year two? Well it's going to be worth what it was in year one times 75% again. So it's going to be $600 times 75% times 75%. And so you could write that as times 75% squared. And I think you see a pattern. In general, if we have gone, let's just call it t years, well then the value of our phone, if we're saying it in dollars, is just going to be $600 times, and I could write it as a decimal, 0.75, instead of 75%, to the t power. So V of t is going to be equal to 600 times 0.75 to the t power, and we're done. Let's do another example. So here, we are told that a biologist has a sample of 6,000 cells. The biologist introduces a virus that kills 1/3 of the cells every week. Write a function that gives the number of cells remaining, which would be C of t, the cells as a function of time, in the sample t weeks after the virus is introduced. So again, pause this video and see if you can figure that out. All right, so I'll set up another table again. So this is time, it's in weeks, and this is the number of cells, C. We could say it's a function of time. So time t equals zero, when zero weeks have gone by, we have 6,000 cells. That's pretty clear. Now after one week, how many cells do we have? What's C of one? Well it says that the virus kills 1/3 of the cells every week, which is another way of saying that 2/3 of the cells are able to live for the next week. And so after one week, we're going to have 6,000 times 2/3. And then after two weeks, or another week goes by, we're gonna have 2/3 of the number that we had after one week. So we're gonna have 6,000 times 2/3 times 2/3, or we could just write that as 2/3 squared. So once again, you are likely seeing the pattern here. We are going to, at time t equals zero, we have 6,000, and then we're going to multiply by 2/3 however many weeks have gone by. So the cells as a function of the weeks of t, which is in weeks, is going to be our original amount, and then however many weeks have gone by, we're going to multiply by 2/3 that many times, so times 2/3 to the t power. And we're done.