A garden has a carrying capacity of 200 trees and needs a rate of three percent per month to be full grown. If the initial number of full-grown trees in the garden is 20 trees, what will the number of full-grown trees be after nine months?
In this question, we’re given some information about a garden and the number of full-grown trees in this garden. We’re told that the carrying capacity of the garden will be 200 trees. And we’re also told information about the growth rate. We’re told the growth rate is three percent per month. We’re also told information about the initial number of fully grown trees in our garden. We’re told that this is equal to 20 trees. We need to use this information to find the number of fully grown trees in our garden after nine months.
Since we’re talking about the population of trees in a garden with a carrying capacity and a growth rate, we know we can model this by using a logistic growth model. So let’s recall the logistic growth model on a population 𝑃 with time 𝑡. This tells us that d𝑃 by d𝑡 will be equal to 𝐾 times 𝑃 multiplied by one minus 𝑃 divided by 𝐿, where 𝐾 is the growth rate of our population and 𝐿 is the carrying capacity of our population.
So in our case, we’ll call the population of fully grown trees in our garden 𝑃 of 𝑡. And it’s also worth pointing out that we notice that our value of the growth rate 𝐾, three percent, is given per month. So when we use this value of 𝐾, that means the units of time 𝑡 will also be in per month, which means when we try to use the logistic growth model to approximate the population after nine months, we’ll need to substitute 𝑡 is equal to nine.
So we could substitute the values of 𝐾 and 𝐿 into our logistic growth model and then solve this by using what we know about separable differential equations. However, this is not necessary because we already know the general solution to the logistic growth model. The general solution to the logistic growth model is as follows. 𝑃 of 𝑡 is equal to 𝐿 divided by one plus 𝐴 times 𝑒 to the power of negative 𝐾𝑡, where 𝐴 is equal to 𝐿 minus 𝑃 of zero all divided by 𝑃 of zero. And 𝑃 of zero is the population when 𝑡 is equal to zero. In other words, it’s the initial population.
And we can also see we’re given the initial population in this question. We’re told that initially there are 20 fully grown trees in our garden. So we’re now almost ready to use our logistic growth model. First, we know 𝑃 of zero is equal to 20. Next, we know our growth rate is three percent. And we need to write this as a decimal. We’ll write this as 0.03. Finally, we’re told the carrying capacity 𝐿 of our garden is 200 trees. So our value of 𝐿 is 200. We’re now ready to start finding our solution.
Let’s start by finding the value of 𝐴. That’s 𝐿 minus 𝑃 zero all divided by 𝑃 zero. So we substitute 𝐿 is 200 and 𝑃 of zero is 20. This gives us 200 minus 20 all divided by 20. And if we calculate this, we get nine. Now that we know the value of our carrying capacity, our growth rate, and the value of 𝐿, we can find a formula for 𝑃 of 𝑡. Substituting in our carrying capacity 𝐿 is equal to 200, our growth rate 𝐾 is 0.03, and our value of the constant 𝐴 is equal to nine, we get that the population of trees in our garden after 𝑡 months is equal to 200 divided by one plus nine times 𝑒 to the power of negative 0.03𝑡.
But remember, the question wants us to find the population of trees after nine months. So we need to substitute 𝑡 is equal to nine into this equation. Substituting in 𝑡 is equal to nine, we get the population of trees in our garden after nine months is equal to 200 divided by one plus nine times 𝑒 to the power of negative 0.03 multiplied by nine. And if we evaluate this expression to one decimal place, we get 25.4. So we can round this down and get 25 trees, giving us our final answer of 25 trees.
In this question, we were given a real-world problem involving the number of fully grown trees in a garden. We were able to convert this information into a model helping us estimate the number of fully grown trees in the garden after nine months. We found that there would be 25 fully grown trees in our garden after nine months.