Video: Estimating How Long It Takes a Population to Grow by Using the Logistic Growth Model

A population is growing following the logistic growth model with an initial population of 7,700 and then a population of 8,500 at 𝑑 = 2 years. Assuming a π‘˜-value of 0.05, after how many years will the population reach 100,000? Round your answer to the nearest whole number of years for durations over 10 years and to one decimal place for durations under 10 years.

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Video Transcript

A population is growing following the logistic growth model with an initial population of 7,700 and then a population of 8,500 at 𝑑 is equal to two years. Assuming a π‘˜-value of 0.05, after how many years will the population reach 100,000? Round your answer to the nearest whole number of years for durations over 10 years and to one decimal place for durations under 10 years.

The question tells us we have a population which follows the logistic growth model. And we’re told in this logistic growth model, the initial population is 7,700. And after 𝑑 is equal to two years, the population reaches 8,500. We’re also told that our model will have a π‘˜-value of 0.05. We need to use this information to find out after how many years our model will predict the population will reach 100,000. And depending on this value, we need to round our answer differently. If this number is over 10 years, we need to round our answer to the nearest whole number. However, if this number is less than 10 years, we need to round it to one decimal place.

To start, let’s call our population after 𝑑 years as predicted by our logistic growth model 𝑃 of 𝑑. Then, the question is asking us which value of 𝑑 will give us that 𝑃 of 𝑑 is equal to 100,000. Since 𝑃 of 𝑑 is given by our logistic growth model, let’s start by recalling what a logistic growth model is. The logistic growth model tells us that we can model our population by the following differential equation. d𝑃 by d𝑑 is equal to π‘˜ times 𝑃 multiplied by one minus 𝑃 divided by 𝐿, where π‘˜ is a measure of the growth rate of our population and 𝐿 is the carrying capacity.

And we could solve this differential equation by using partial fractions. However, we also know a formula for the solution of this differential equation. We can find the general solution 𝑃 of 𝑑 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative π‘˜π‘‘. And this will be a solution for any value of the constant 𝐴. And one way to find the specific solution to find the value of 𝐴 is to use the fact that 𝐴 is equal to the carrying capacity minus the initial population all divided by the initial population. So, let’s see what’s information we’re given in the question.

First, we’re told the initial population is equal to 7,700. So, 𝑃 of zero is equal to 7,700. Next, we’re told when 𝑑 is equal to two years, our population will be 8,500. So, 𝑃 of two is equal to 8,500. And it’s worth reiterating at this point, we’re measuring our value of 𝑑 in years. Next, we’re also told the value of π‘˜, our growth rate, is equal to 0.05. And this is all the information we’re given. We can see we’re not told the value of the carrying capacity. So, we need to use this information to find the specific solution to our differential equation 𝑃 of 𝑑.

To start, we’re told the value of 𝑃 of zero and we’re told the value of 𝑃 of two. The first thing we’ll do is substitute 𝑑 is equal to zero into our expression for 𝑃 of 𝑑. And since we know that 𝑃 of zero is 7,700, this gives us 7,700 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 0.05 multiplied by zero. First, 𝑒 to the power of negative 0.05 multiplied by zero is just equal to 𝑒 to the zeroth power, which is, of course, just equal to one. So, we can simplify our denominator to just be equal to one plus 𝐴.

And we’ll do one more thing before we continue. We’ll multiply both sides of this equation through by our denominator one plus 𝐴. This gives us that 7,700 multiplied by one plus 𝐴 is equal to 𝐿. Next, we’re going to substitute 𝑑 is equal to two into our expression for 𝑃 of 𝑑. So, substituting 𝑑 is equal to two and using the fact that 𝑃 of two is equal to 8,500, we get that 8,500 is equal to 𝐿 divided by one plus 𝐴𝑒 to the power of negative 0.05 times two.

And now, we can see we have two equations for 𝐿 and 𝐴. So, we can solve these simultaneous equations to find the values of 𝐿 and 𝐴. We’ll do this by rearranging both of our equations to make 𝐿 the subject. We’ve already done this for the first equation. So, let’s now do this for our second equation. We can see 𝑒 to the power of negative 0.05 times two is equal to 𝑒 to the power of negative 0.1. So, we can simplify our second equation to give us 8,500 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 0.1. We’ll rearrange this by multiplying both sides of our equation through by the denominator. This gives us 8,500 multiplied by one plus 𝐴𝑒 to the power of negative 0.1 is equal to 𝐿.

So, now, we have two equations which are both equal to 𝐿. So these two equations must be equal to each other. So, setting these two equations equal to each other, we get 7,700 times one plus 𝐴 is equal to 8,500 multiplied by one plus 𝐴𝑒 to the power of negative 0.1. We want to solve both of these equations for 𝐴. So, we’ll distribute both of our coefficients over both sets of parentheses. Distributing our first set of parentheses, we get 7,700 plus 7,700𝐴. And distributing 8,500 over our second set of parentheses gives us 8,500 plus 8,500𝐴𝑒 to the power of negative 0.1.

We want to solve this equation for 𝐴. To do this, we’ll start by subtracting 8,500 from both sides of the equation. And we can then simplify this expression. We see negative 8,500 plus 7,700 is equal to negative 900. To solve this equation for 𝐴, we want all of our variables 𝐴 on the same side of this equation. So, we’ll subtract 7,700𝐴 from both sides of the equation. Then, all we need to do is factor 𝐴 from the right-hand side of our equation. So, by taking out a factor of 𝐴 from the right-hand side of our equation, we now have negative 900 is equal to 𝐴 times 8,500𝑒 to the power of negative 0.1 minus 7,700.

And now, we can just solve this equation for 𝐴. We just need to divide through by the coefficient of 𝐴. Doing this, we get that 𝐴 is equal to negative 900 divided by 8,500𝑒 to the power of negative 0.1 minus 7,700. And if we calculate this, we get approximately 101.329. We’ll use the exact value of 𝐴 in our future calculations.

Let’s take a closer look at our solution 𝑃 of 𝑑. We found the value of 𝐴, and we know the value of π‘˜. So, all we need to do is find the value of 𝐿. And we see we can find the value of 𝐿 from the value of 𝐴 and our initial population. And we know both of these values. So, let’s clear some space and find the value of our carrying capacity 𝐿.

To start, we know that 𝐴 is equal to 𝐿 minus 𝑃 of zero all divided by 𝑃 of zero. And we know the values of 𝐴 and 𝑃 of zero. So, we just need to rearrange this equation for 𝐿. We’ll start by multiplying through by 𝑃 of zero. This gives us 𝐴 times 𝑃 of zero is equal to 𝐿 minus 𝑃 of zero. We’ll solve this equation by adding 𝑃 of zero to both sides of the equation. Then, we can just solve this equation for 𝐿 by substituting in 𝑃 of zero is 7,700 and our expression for our value of 𝐴.

This gives us the following expression for our value of 𝐿. And we could calculate this exactly. However, the question only needs us to give at most one decimal place of accuracy. So, we’ll give this to the nearest integer is 787,934. So, in our solution, we can now see we have expressions for 𝐴, we know the value of π‘˜, and we found the value of 𝐿. So, we can find our solution 𝑃 of 𝑑. So let’s clear some space and write our solution 𝑃 of 𝑑.

Using our approximations for 𝐴 and 𝐿 and using our value of π‘˜, we get that 𝑃 of 𝑑 is approximately equal to 787,934 divided by one plus 101.329 multiplied by 𝑒 to the power of negative 0.05𝑑. Remember, the question wants us to find the value of 𝑑, where 𝑃 of 𝑑 is equal to 100,000. So, we need to solve this equation is equal to 100,000 for 𝑑. We’ll start by multiplying both sides of the equation through by our denominator. This gives us the following expression. Next, we’ll divide both sides of our equation through by 100,000. And of course, dividing 787,934 by 100,000 just gives us 7.87934.

Next, to solve this equation for 𝑑, we’ll subtract one from both sides of the equation. And we can just calculate this expression. Next, we want to divide both sides of our equations through by 101.329. This leaves us with the following expression. To solve this equation for 𝑑, we now need to take the natural logarithm, of both sides of the equation. And of course, we know the natural logarithm and exponential function are inverse functions. So, the natural logarithm of 𝑒 to the power of negative 0.05𝑑 is just equal to negative 0.05𝑑.

And finally, we can solve this equation for 𝑑 by dividing both sides of our equation through by negative 0.05. To one decimal place, this gives us 𝑑 is approximately equal to 53.8 years. However, remember, the question tells us to give our answer to the nearest whole number of years for durations over 10 years. So, we need to round our answer up to the nearest whole number, which in this case is 54.

Therefore, by using a logistic growth model, we were able to show after approximately 54 years, this population will reach 100,000.

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