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“This video. We re going to talk about exponential growth and decay problems as it it relates to population growth. So to speak. Now the first equation need to be with is this equation.
Dp divided by dt is equal to k. Times. 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 dp dt that s how fast the population is grown at a given time k. Is the relative growth rate. Mp. Is the size of the population at some time t.
Now for understand is conceptual. Let s say if we have a sample of a thousand bacteria and let s say that in a 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 2000 bacteria counts well since we have twice the amount of bacteria. We should expect it to grow at twice the rate that is at a hundred cells per hour so thus we could see that the rate at which the population grows is proportional to the population shocks.
So if you increase the population size dp dt will increase. Fortunately if you double the population size the rate will double if you triple ate the rate will triple now from that equation. How can we devise 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 d p. Is equal to k p. Times. D p.
Now we need to separate the variables. Ks are constant. As you said is the relative girlfri. We need to separate p from dt.
So let s divide both sides by p. So now we have this expression. 1 over p. Dp is equal to k dt.
So at this point. Let s integrate both sides. The antiderivative of 1 over p. Is simply the natural log of p k.
Is a constant the antiderivative of dt is t. So this is going to be k times t. Plus. The general constant c.
So what should we do at this point. Now that we have this equation. How can we get t by itself. At this point.
You want to put the equation on top of e. That is on the x alone. Even if ln t is equal to kt plus c. And then e raised to the ln p.
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 to the kt plus c. Now. Just review. Some things in algebra.
You know that x. Squared. Times. X.
Cubed is equal to x to the fifth. We need to do is you need to add the two exponents so what if we work backwards. We can say that x raised to the 4 plus. 7.
Is equal to x to the fourth times x to the seventh 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 h.
To c. Is a constant. 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 of 0. So let s replace t with 0.
This is going to be c e k. Times 0 k times 0. Is simply 0. So this is going to be c e.
To the 0. Now anything raised to the 0 power is equal to 1. So therefore p of 0 is c. Times.
1 so c is the initial value. It s the population at t equals. 0. So we could say c is basically p initial thus we can replace c of t.
Initial. So the general equation is the population at any time t is equal to the initial population t. 0. E.
Times 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 route of 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 of 0.
E raised to the k t. Our goal is to solve for king. But first we got to find p0 so what exactly is p0 well p of 0 is equal to 1500 when t is 0. That is the year 2000.
The population is 1500 so if we replace p of t with 1500 and t was zero. We re going to get 1500 is equal to p0 e 0. E. To the 0 is 1 so p0 is 1500 so p of 0 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 a value of p of 0. We can say that p of t is 1500 that s p0 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 15 77. So let s replace t of t. With 1577 unless you place t with 1 in order to solve for k. Let s divide both sides by 1515 77.
Divided by 1500 is equal to one point zero. Five. One three. 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 exporting 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 one point zero five one three which is about point zero five. So that s the value of k.
So now we can write a general form so to write the general formula is going to be 1500s e. And then all i need to do is replace k. With point zero five times t. So that s the answer to part b.
That s the general equation that will give you the population p of t. At any time t. The answer to part a the relative growth rate is the value of k. Which is point zero five.
So basically the population increases by five percent every year compounding continuously this formula is the general form 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 point zero five times 10.
And just type it in your calculator. Exactly the way you see a point zero five times ten is point five. So it s 1500 times e raised to the point five. And so.
The population is going to be about two thousand four hundred and seventy three rabbits and 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 times two is three thousand so let s replace p of t. With three thousand. We just gotta 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.
Divided by 1500 is simply 2 so 2 is equal to e raised to the point 0 5. Times 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 of 2 is equal to 005. Times t. Multiplied by the natural log of e.
And the natural log of e is 1. So ln. 2. Is equal to point zero five times t.
So the time it takes for it to double is simply ln two divided by the rate constant k. The relative growth rate. So if we divide these two numbers ln twos like point six nine three one divided by point zero five. So it s going to take thirteen point eight six years in order for the population to double let s round that to the nearest whole number so approximately 14 years or we could say by the year.
2014 the population will be above 3000. And so that s it for this video thanks for ” ..
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