### Video Transcript

It is assumed that the snail population in a kitchen garden can be modeled with a logistic growth model. At π‘ is equal to zero, there were six snails in the patch. One month later, there were 12 snails. Given that the carrying capacity of the kitchen garden is 36 snails, at what time π‘ in months does the model predict a snail population of 32? Round your answer to the nearest whole number of months.

In this question, weβre told that a snail population in a kitchen garden can be modeled by using a logistic growth model. And weβre told when π‘ is equal to zero, there were six snails in the patch. This means our initial population is six. Weβre also told that one month later, so when π‘ is equal to one, there are 12 snails. Weβre also told the carrying capacity for our logistic model; itβs equal to 36. We need to find the times π‘ in months that our model will predict the snail population will have a population of 32. We then need to round this number to the nearest whole number of months.

To start, letβs call the population of our snails after π‘ months π of π‘. The question tells us we can approximate π of π‘ by using a logistic growth model. And it wants us to find the value of π‘ where our model will predict a snail population of 32. Letβs start by recalling logistic growth model. This tells us dπ by dπ‘ is equal to π times π multiplied by one minus π divided by πΏ, where π is a measure of our growth rate and πΏ is the carrying capacity. Weβre told in this case, we can approximate our snail population by using this logistic growth model. So we could start finding values for π and πΏ and then solving this logistic growth model.

However, we already know the solutions to the logistic growth model. We know we can solve the logistic growth model with π of π‘ is equal to πΏ divided by one plus π΄ times π to the power of negative ππ‘, where π΄ is equal to πΏ minus our initial population all divided by the initial population. So the question wants us to find out when our logistic model will approximate the population of the snails will be equal to 32. So we need to find our specific solution to π of π‘. And to do that, we need the value of π΄. So letβs see what information weβre given in the question.

Weβre told when π‘ is equal to zero, there are six snails in the kitchen. So, the initial population of snails is equal to six. So π of zero is equal to six. Next, weβre told one month later, there are 12 snails. This means that π evaluated at one is equal to 12. Itβs worth reiterating weβre using months as our unit of time since the question wants us to use months as the unit of time. Next, weβre told the carrying capacity of the kitchen garden is equal to 36 snails. So our value of πΏ is equal to 36. But now, it seems like we have a problem. We know the value of our carrying capacity and we know the initial population, so we can calculate our value of π΄. So in our solution, we can calculate the value of π΄ and we know the value of πΏ. However, we donβt know the value of π.

However, we do know two values of π of π‘, so we can use these to find the value of π. Letβs start by finding the value of π΄. Itβs equal to the carrying capacity minus the initial population divided by the initial population. In this case, that is 36 minus six all divided by six, which we can calculate is equal to five. So by using this value of π΄ and that the carrying capacity is 36, by our the logistic growth model, we can approximate π of π‘ as 36 divided by one plus five times π to the power of negative ππ‘.

Now, if we were to substitute π‘ is equal to zero into this equation, we would not be able to find the value of π since we would then be multiplying π by zero. Instead, we want to substitute π‘ is equal to one since we know π of one is equal to 12. So weβll substitute π‘ is equal to one into this equation. This gives us π of one, which we know is 12, is equal to 36 divided by one plus five times π to the power of negative π multiplied by one. Of course, multiplying negative π by one is just equal to negative π. Now, we want to solve this equation for π. Weβll start by multiplying both sides of this equation through by our denominator one plus five π to the power of negative π. This gives us 12 times one plus five π to the power of negative π is equal to 36.

We can simplify this further. Weβll divide both sides of this equation through by 12. And of course, we can calculate 36 divided by 12 to be equal to three. So we now have one plus five π to the power of negative π is equal to three. Weβll subtract one from both sides of this equation. This gives us three minus one, which is of course equal to two. Remember, we want to solve this equation for π, so weβll divide through by five. This gives us π to the power of negative π is equal to two divided by five, which weβll write as a decimal, 0.4.

Now, we want to solve the equation π to the power of negative π is equal to 0.4. Weβll do this by taking the natural logarithm of both sides of the equation. This gives us the natural logarithm of π to the power of negative π is equal to the natural logarithm of 0.4. But remember, the natural logarithm and the exponential function are inverse functions. So taking the natural logarithm of π to the power of negative π is just equal to negative π. Finally, weβll multiply both sides of this equation by negative one. We found that π is equal to negative the natural logarithm of 0.4.

Now, we know the value of our carrying capacity πΏ, we calculated the value of π΄, and we found the growth rate π. So we can find our expression for the snail population by using the logistic growth model. Substituting in our values for πΏ, π΄, and π, we get that π of π‘ is equal to 36 divided by one plus five π to the power of negative one times negative the natural logarithm of 0.4π‘. And we can simplify this. In our exponent of π, we have negative one times negative one is equal to one. This gives us 36 divided by one plus five π to the power of the natural logarithm of 0.4 times π‘.

And we can simplify this further. We can notice π to the power of the natural logarithm of 0.4 times π‘ is actually equal to π to the power of the natural logarithm of 0.4 all raised to the power of π‘. And then remember, the exponential function and natural logarithm function are inverse functions. So we can simplify this to just give us 0.4 raised to the power of π‘. So this gives us our model of π of π‘ is equal to 36 divided by one plus five times 0.4 raised to the power of π‘. The question wants us to find the value of π‘ where this is equal to 32. So letβs clear some space and solve for this value of π‘.

Weβll start by multiplying both sides of the equation through by our denominator. Weβll multiply by one plus five times 0.4 raised to the power of π‘. This gives us 36 is equal to 32 times one plus five times 0.4 raised to the power of π‘. Now, weβll divide both sides of the equation through by 32. And of course, we can simplify 36 divided by 32 to give us nine divided by eight. So we now have nine over eight is equal to one plus five times 0.4 raised to the power of π‘. Weβll subtract one from both sides of this equation. This gives us one-eighth is equal to five times 0.4 raised to the power of π‘. Weβll solve this by dividing both sides of the equation through by five. This gives us one over 40 is equal to 0.4 raised to the power of π‘.

Weβll solve this by taking logs of both sides of the equation. Taking the natural logarithm of both sides of the equation, we get the natural logarithm of one over 40 is equal to the natural logarithm of 0.4 raised to the power of π‘. And weβll simplify the right-hand side of this equation by using our power rule for logarithms. The natural logarithm of 0.4 raised to the power of π‘ is equal to π‘ times the natural logarithm of 0.4. Now, all we have to do is divide both sides of the equation by the natural logarithm of 0.4. And this gives us our value of π‘ is the natural logarithm of one over 40 divided by the natural logarithm of 0.4.

And if we calculate this expression and write this to the nearest whole number of months, we get that the answer is four. Therefore, by using a logistic growth model, we were able to show the snail population in this kitchen garden will be at 32 in approximately four months.