Transcription
In this video, we're going to talk about exponential growth and decay problems as it relates to, uh, population growth, so to speak. Now, the first equation you need to be familiar with is this equation: dP/dT = K * P. This equation tells us that the population grows at a rate that is proportional to the size of the population. The population growth rate is dP/dt; that's how fast the population is growing at a given time. K is the relative growth rate, and P is the size of the population at some time T.
Now, to understand this conceptually, let's say if we have a sample of a, uh, bacteria, and let's say that in this sample the bacteria grows 50 cells per hour. What's going to happen if we double the population? Let's say if we have a sample of 2,000 bacteria counts. Well, since we have twice the amount of bacteria, we should expect it to grow at twice the rate, that is, at 100 cells per hour. So thus, we could see that the rate at which the population grows is proportional to the population size. So if you increase the population size, dP/dt will increase proportionally. If you double the population size, the rate will double; if you triple it, the rate will triple.
Now, from that equation, how can we derive a general formula to calculate the population at any time t? In order to do this, let's multiply both sides by dt, so that these two will cancel, and so we have dP = K * P * dt. Now we need to separate the variables. K is a constant, as we said, it's the relative growth rate. We need to separate P from dt, so let's divide both sides by P. So now we have this expression: (1/P) dP = K dt. So at this point, let's, uh, integrate both sides. The anti-derivative of 1/P is simply the natural log of P. K is a constant; the anti-derivative of dt is t, so this is going to be K * t plus the general constant C. So what should we do at this point? Now that we have this equation, how can we get P by itself? At this point, you want to put the equation on top of e, that is, on the exponent of e. If ln P = Kt + C, then e raised to the ln P is equal to e raised to the Kt + C. Now, the base of a natural log is base e, so these two will cancel, and so P is equal to e raised to the Kt + C.
Now, just review some things in algebra. You know that x² * x³ = x⁵. What you need to do is you need to add the two exponents. So what if we work backwards? We could say that x raised to the 4 + 7 is equal to x⁴ * x⁷. Basically, we're starting here and going back to this form. Notice that Kt is added to C, so therefore what we could do is we could say that e to the Kt + C is equal to e raised to the Kt times e to the C. Now, e to the C is a constant, so we could simply replace it with C. So the population is equal to Ce to the Kt.
Now, starting with this equation, we're going to calculate P(0). So let's replace t with zero. This is going to be Ce^(K*0). K * 0 is simply zero, so this is going to be Ce⁰. Now, anything raised to the zero power is equal to 1, so therefore P(0) = C * 1. So C is the initial value; it's the population at t = 0. So we could say C is basically P initial. That's why we can replace C with P initial. So the general equation is: the population at any time t is equal to the initial population, P₀, e to the (relative growth rate, which is K, multiplied by t). So now you know how to derive this formula from this expression.
Let's work on this problem. The table below shows the rabbit population on a certain island, where T is the number of years beginning with the year 2000. Determine the relative growth rate. So we need to use the equation P(t) = P(0)e^(Kt). Our goal is to solve for K, but first we got to find P₀. So what exactly is P₀? Well, P(0) is equal to 1500 when t is zero; that is, the year 2000, the population is 1500. So if we replace P(t) with 1500 and t with 0, we're going to get 1500 = P₀e⁰. e⁰ is 1, so P₀ is 1500. So P(0) is the same as P₀; that's the initial population. So how can we use this to calculate K? So now that we have the value of P₀, we can say that P(t) = 1500e^(Kt). So now let's use another point to calculate K. We can use any point in the data table, but let's use the first one. In the year 2001, t is equal to 1, so the population at t = 1 is, uh, 1577. So let's replace P(t) with 1577 and let's replace t with one in order to solve for K. Let's divide both sides by 1500. 1577/1500 is equal to 1.0513, and that's equal to e^K. So now at this point, we need to take the natural log of both sides. The natural log of e^K, what do you think that's equal to? A property of logs allows us to move the constant or the exponent to the front, so this is equal to K times the natural log of e. The natural log of e is equal to 1, so K is simply equal to ln 1.0513, which is about 0.05. So that's the value of K.
So now we can write a general formula. So to write the general formula, it's going to be 1500e, and then all we need to do is replace K with 0.05 * t. So that's the answer to Part B; that's the general equation that will give you the population P(t) at any time t. The answer to Part A, the relative growth rate, is the value of K, which is 0.05. So basically, the population increases by 5% every year, compounding continuously. This formula is the general formula of the compound interest type problems.
Now, how can we estimate the population in 2010? To do that, simply replace t with 10. So this is going to be 1500e^(0.05 * 10), and, and just type it in your calculator exactly the way you see it. 0.05 * 10 is 0.5, so it's 1500 * e^0.5, and so the population is going to be about 2,473 rabbits in the year 2010.
Now, how many years will it take the population to double? And that's starting from the year 2000. What we're going to do is we're going to find out how long it takes for the population to double from 1500 to 3000, because 1500 * 2 is 3000. So let's replace P(t) with 3000, and we just got to solve for the variable t. So let's get rid of a few things. So our first step is to divide both sides by 1500. 3000/1500 is simply 2, so 2 = e^(0.05 * t). Next, we need to take the natural log of both sides. This will allow us to take this exponent and move it to the front, so the natural log 2 is equal to 0.05 * t times the natural log of e, and the natural log of e is 1. So ln 2 = 0.05 * t. So the time it takes for it to double is simply ln 2 divided by the rate constant K, the relative growth rate. So if we divide these two numbers, ln 2 is like 0.6931/0.05, so it's going to take 13.86 years. This video, thanks for watching.