 Many phenomena can be modeled using an exponential function. For example, the concentration of medication in the bloodstream, the amount of a radioactive substance remaining, or the number of infected persons in an epidemic. For example, one very common use of exponential functions involves something called the half-life. The half-life of a substance is the amount of time it takes for half of a substance to break down. Suppose plastic bottles in a landfill have a half-life of 450 years. If 1,000 bottles are dumped into a landfill, how many will be left after 100 years, and how long will it take before 90% of them decompose? Now because we are given a half-life, we'll assume the number N of plastic bottles is modeled using an exponential function N equals E to power kT, where A and k are unknown values. Now if we start, that's T equals 0, with 1,000 bottles, we have N equals 1,000 when T equals 0, so we can write the equation 1,000 equals AE to power k times 0. And we can simplify a little bit and solve for A. And so we find the number of plastic bags after T years to be N equals 1,000 E to power something. Since the half-life is given as 450 years, that tells us that if T equals 450, the number is going to be half, 1,000 divided by 2 or 500. And so this gives us another equation. And we can solve that for the unknown value of k, which works out to be, and that gives us our function that tells us the number of bottles after T years. After T equals 100 years, we'll let T equals 100 and substitute it in, and we find that we have 857 bottles remaining. Now if we want 90% of the bottles to have decomposed, then there will be 10% remaining. Well, 10% of 1,000 is 100, and so we want to solve for T when N is equal to 100. So substituting these values into our equation, and solving, and so it'll take almost 1,500 years for 90% of the bottles to be decomposed. Or we could consider a totally unrealistic case, an irresponsible demagogue denies the existence of a pandemic, but later record provide the following data. Assuming the number of cases can be modeled by an exponential function, how long will it take before the number of cases doubles? How long will it be before there are 1,000 cases? And how long before the irresponsible demagogue is held accountable? So again, we're assuming an exponential model, and we have at 50 days 137,000 cases, and at 60 days 195,000 cases. And we can substitute those into our function. And we have two equations with two unknowns, and there are several ways to solve this system. However, the easiest thing to do is notice that since both of these have a factor of A, if we divide one by the other, and simplify, we get an exponential equation that we could then solve for K, which gives us part of our exponential model. Now we can also solve for A, because we know that at day 60, there's 195,000 cases. And since we have K, we can substitute these into our model and solve for A. Now we want, well actually we don't want, but the question is asking us to find when the number of cases doubles. So here on day 60, we have 195,000 cases, and so there will be twice as many, 390,000 cases when, and if we solve that, we find T is about equal to 80. Now it's important to recognize here is that it's already day 60. So that means the number of cases will double just 20 days later. How about when there's going to be 1,000,000 cases? So that's 1,000,000 cases, and we have our model, and so we find about day 106, which is in 46 days. As for the last question, in a perfect world, the irresponsible demagogue will be held accountable. We don't live in that world, but we should work towards it.