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 is equal to K * P so 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 dpdt 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 th bacteria and let's say that in this sample the B Bia 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 of 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 dpdt 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 te in order to do this let's multiply both sides by DT so that these two will cancel and so we have DP is equal to k p * DT now we need to separate the variables K is a constant as we said it's the relative growth frame we need to separate P from DT so let's divide both sides by P so now we have this expression 1/ P DP is equal to 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 PP is equal to KT plus C then e raised to the LMP is equal to e raised to the KT plus C now the base of a natural log is base e so these two will cancel and so p is equal to e raised the KT plus C now just review some things in algebra you know that x^2 * X Cub is equal to X 5th 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 4 + 7 is equal to x 4 4 * X 7th 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 plus C is equal to e raised to the KT times e to the C now e to the C is a Conant so we could simply replace it with C so the population is equal to c e to the KT now starting with this equation we're going to calculate P of0 so let's replace t with zero this is going to be c e K * 0 K * 0 is simply zero so this is going to be c e to 0 now anything raised to the Z power is equal to 1 so therefore P of 0 is C * 1 so C is the initial value it's the population at T equals z so we could say C is basically P initial that's we can replace C with P initial so the general equation is the population at any time T is equal to the initial population p 0 e time the relative growth rate which is K multipli 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 pop 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 of T is equal to p of0 e raised to the KT our goal is to solve for K but first we got to find p 0 so what exactly is P0 well p of 0 is equal to 1500 when T is Zer that is the year 2000 the population is 1500 so if we replace P of t with 1500 and T with 0 we're going to get 1500 is equal to p 0 e to0 e to0 is 1 so P0 is 1500 so p of0 is the same as P Sub 0 that's the initial population so how can we use this to calculate K so now that we have the value of P of0 we can say that P of t is 1500 that's p 0 e 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 equal 1 is uh 1577 so let's replace P of t with 1577 and let's replace t with one in order to solve for K let's divide both sides by 1500 1577 ID 1500 is equal to 1.05 13 and that's equal to e raised to the K power so now at this point we need to take the natural log of both sides the natural log of e to the 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 time the natural log of e the natural log of e is equal to 1 so K is simply equal to Ln 1.05 13 which is about 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 1500 e and then all we need to do is replace K with 05 time T so that's the answer to Part B that's the general equation that will give you the population P oft at any time T the answer to part A the relative growth rate is the value of K which is 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 1500 e raised to the 05 * 10 and and just type it in your calculator exactly the way you see it 05 * 10 is .5 so it's 1500 * e raised to the .5 and so the population is going to be about 2,473 Rabbids 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 3,000 because 1500 * 2 is 3,000 so let's replace P of t with 3,000 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 3,000 ID 1500 is simply 2 so 2 is equal to e raed to the 05 * T next we need to take the natural log of both sides this will allow us to take this exponent move to the front so the natural log 2 is equal to 05 * ttip the natural log of e and the natural log of e is 1 so Ln 2 is equal to 05 * T so the time it takes for it to double is simply ln2 divided by the rate constant K the relative grow rame so if we divide these two numbers I two is like 6931 ID 05 so it's going to take 13.86% this video thanks for watching