97–100. Logistic growth Scientists often use the logistic grow...

Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model.

{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.

b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says in a newly established wildlife reserve, 100 rabbits are introduced into an area with an estimated carrying capacity of 10,000 rabbits. A logistic model for the rabbit population is given by RFT is equal to 1 million divided by the quantity of 100 plus 9900 multiplied by E raised to the quantity of minus 0.3 multiplied by T. Where T here is measured in years, determine the time when the population reaches 6000 rabbits. Also find the time when the population reaches 90% of the care capacity. And we need to round our final answer to the nearest integer. Now this question wants us to find the time at which the population reaches 6000 rabbits, as well as when the population reaches 90% of the carrying capacity. So we're gonna need to take our function RFT and solve it for T in order to determine the time. So starting with our function RFT. Which is equal to 1 million. Divided by the quantity. Of 100. Plus 9900 multiplied by E raises to the quantity of minus 0.3, multiplied by T. So, we need to solve this for T. So, our first step is to multiply both sides of the equation by the denominator of a fraction on the right hand side, and this will remove the fraction out of our equation. So doing so, we'll have RFT. Multiplied by the quantity of 100 plus 9900, multiplied by E raised to the quantity of minus 0.3, multiplied by T in quantity, and this is going to be equal to 1 million. So, our next step here is to distribute our RFT and doing so, we will get 100 multiplied by RFT. Plus 9900 Multiplied by R of T. Multiplied by E raised to the quantity of minus 0.3, multiplied by T in quantity, and this is equal to 1 million still. So now we need to subtract 100 multiplied by RT from both sides of the equation. Doing so, we will have 9900 multiplied by RFT. Multiplied by E raised to the quantity of minus 0.3, multiplied by T in quantity. This is going to be equal to 1 million. Minus 100 multiplied by RFT. So, the next step is to divide both sides of the equation by 9900 multiplied by RT. In doing so, we will have E raised to the quantity of minus 0.3, multiplied by T. and this is going to be equal to the quantity of 1 million. Minus 100 multiplied by RFT in quantity, divided by the quantity of 9900 multiplied by RFT in quantity. So to solve for T, we need to take the natural log of both sides of this equation to get rid of our exponential. So taking the natural log of both sides of this equation, we will get minus 0.3, multiplied by T. is equal to The natural log of the quantity and here for a fraction in the numerator will have the quantity of 1 million. Minus 100 multiplied by RFT. In quantity divided by the quantity of 9900 multiplied by RFT. In quantity So to solve for T, we now just need to divide both sides by minus 0.3. In doing so, we will have T is equal to -1 divided by 0.3. That is multiplied by the natural log of the quantity, and again here for a fraction inside the natural log function, we will have in the numerator the quantity of 1 million. -100 multiplied by RFT. In quantity, and that quantity is divided by the quantity of 9900 multiplied by RFT. In quantity So now we need to determine the time when our population is equal to 6000. That means we will set RFT. Equal to 6000. And substitute in that value into our expression for tea, we will have tea. is equal to -1 divided by 0.3. Multiplied by the natural log of the quantity, and in the numerator we have the quantity of 1 million. Minus 100 Multiplied by 6000. In quantity, and that quantity is divided by the quantity of 9900 multiplied by 6000. In quantity And when we evaluate this expression in our calculator and round to the nearest integer, this is approximately equal to 17 years. So this is the first half of our question answered. Now, for the second half, we need to determine the time at which the population reaches 90% of the carrying capacity. So that means that RFT It's going to be equal to 90% or 0.9%, multiplied by the carrying capacity, which is 10,000. And this is going to be equal to 9000. So, RFT is going to be equal to 9000 in the second case. So substituting in that value into our expression for T, we will have T is equal to minus 1 divided by 0.3, multiplied by the natural log of the quantity, and here again in a fraction, in the numerator, we have the quantity of 1 million. Minus 100 Multiplied by 9000. In quantity, and that quantity is divided by the quantity of 9900 multiplied by 9000. In quantity And again, using our calculator to evaluate this expression and rounding to the nearest integer, this is approximately equal to 23 years. So that is the answer to the second half of our problem. So, our final answer for this problem is that To reach the population of 6000 rabbits, it takes 17 years. And to reach 90% of the carrying capacity, it takes 23 years. So I hope that this explanation has been useful, and I'll see you in the next video.

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