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- [Voiceover] Chepi is an ecologist who studies the change in the narwhal population of the Arctic ocean over time. She observed that the population population loses 5.6% of its size every 2.8 months. The population of narwhals can be modeled by a function, N, which depends on the amount of time, t in months. When Chepi began the study, she observed that there were 89,000 narwhals in the Arctic ocean. Write a function that models the population of narwhals t months since the beginning of Chepi's study. Like always, pause the video and see if you can do it on your own before we work through it together. So let's now work through it together. To get my essence of what this function is to do, it's always valuable to see, to create a table for some interesting inputs for the function and seeing how the function should behave. So first of all, if t is in months and N of t is, in N is the, that models, N is the number of narwhals, the narwhals. So what, when T is equal to zero, what is N of zero? Well, we know a T equals zero. There are 89,000 narwhals in the ocean. So 89,000. And now, what's another interesting one? Well, T is in months and we know that the population decreases 5.6% every 2.8 months. So let's think about when T is 2.8, 2.8 months. Well then the population, it should have gone down 5.6%. So going down 5.6% is the same thing as retaining. What's one minus 5.6%? Retaining 94.4%. I'll be clear. 100%, if you lose 5.6%, you are going to be left with 94.4%. The 0.6 plus 0.4 gets you a 95 plus another five is 100. So another way of saying, this sentence, that the population loses 5.6% of its size every 2.8 months is to say that the population is 94%, 94.4% of its size every 2.8 months or shrinks to 94.4% of its original size every, or let me phrase this clearly. After every 2.8 months, the population, you can either say it shrinks 5.6% or you could say it has, it's gone from, it's 94.4% of the population at the beginning of those 2.8 months. So after 2.8 months, the population should be 89,000 times, I could write times 94.4% or I could write times 0.944. Now, if we go another 2.8 months, so two times 2.8. I have to say you could just write that as, I could write that as 5.6 months but let me just write this is 2.8 months. Where are we gonna be? We're gonna be 89,000 times 0.944. This is where we were before at the beginning of this period. Now, we're gonna be 94.4% of that. So we're gonna multiply by 94.4% again or 0.944 again or we can just say times 0.944 squared. And after three of these periods, well, we're gonna be times 0.944 again. So it's gonna be 89,000 times 0.944 squared times 0.944 which is gonna be 0.944 to the third power. And I think you might see what's going on here. We have an exponential function. Between every 2.8 months, we are multiplying by this common ratio of 0.944. And so we could write our function N of t. Our initial value is 89,000 times 0.944 to the power of, however, many of these 2.8 month period we've gone so far. So if we take the number of months and we divide by 2.8, that's how many 2.8 month period we have gotten gone. And so notice, when t equals zero, all of this turns into one, erasing the zero part, that becomes one. You have 89,000. When t is equal to 2.8, this exponent is one. Now we're gonna multiply by 0.944 once. When t is 5.6, the exponent is going to be two. Now we're gonna multiply by 0.944 twice. And I'm just doing the values that make the exponent integers but it's going to work for the ones in between. I encourage you to graph it or to try those values in a calculator if you like. But there you have it, we're done. We have modeled our narwhals. So let me just underline that and we're done.