Constructing exponential models: percent change | Mathematics II | High School Math | Khan Academy

Cheppy is an ecologist who studies the change in the narwhal population of the Arctic Ocean over time. She observed that the 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 Cheppy 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 Cheppy's study. Like always, pause the video and see if you can do it on your own before we work through it together.

So now let's work through it together. To get my sense of what this function needs to do, it's always valuable to create a table for some interesting inputs for the function and see how the function should behave.

First of all, if t is in months and n of t is the number of narwhals, what happens when t is equal to zero? Well, we know at t equals zero there are 89,000 narwhals in the ocean, so 89,000.

Now, what's another interesting one? We know that the population decreases 5.6% every 2.8 months. Let's think about when t is 2.8 months. The population should have gone down 5.6%.

Going down 5.6% is the same thing as retaining what? What’s one minus 5.6? Retaining 94.4%. Let me be clear: if you lose 5.6%, you are going to be left with 94.4% of the population.

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 94.4% of its size every 2.8 months, or shrinks to 94.4% of its original size every 2.8 months.

So, after 2.8 months, the population should be 89,000 times 94.4%, or we could write that as 89,000 times 0.944. Now, if we go another 2.8 months, two times 2.8—obviously, I could just write that as 5.6 months—but let me just write this as 2.8 months again.

Where are we going to be? We're going to be at 89,000 times 0.944 (this is where we were before at the beginning of this period), and we're going to multiply by 94.4% again, or 0.944 again, or we can just say times 0.944 squared.

After three of these periods, we'll be at 89,000 times 0.944 squared times 0.944, which is going to be 0.944 to the third power. 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. Therefore, we can 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 periods we've gone through.

If we take the number of months and we divide by 2.8, that tells us how many 2.8-month periods we have gone through. Notice that when t equals zero, all of this turns into one; you raise something to the zeroth power, and it just becomes one. We have 89,000.

When t is equal to 2.8, this exponent is one, and we're going to multiply by 0.944 once. When t is 5.6, the exponent is going to be 2, and we're going to multiply by 0.944 twice.

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 on a calculator if you like. But there you have it, we're done; we have modeled our narwhals.

So, let me just underline that—we're done!