Question

A lake is stocked with 500 fish, and their population increases according to the logistic curve p(t)=10,000/1+19e⁻ᵗ/⁵, where t is measured in months. At what rate is the fish population changing at the end of 4 months? Round your answer to one decimal place.

Answer 1

The fish population is changing at a rate of -47.2 fish per month at the end of 4 months.

Step-by-step explanation:

To find the rate at which the fish population is changing at the end of 4 months, we can use the derivative of the logistic function p(t). The derivative represents the rate of change of the population with respect to time. In this case, we need to find the derivative of p(t) and then substitute t=4.

We calculate the derivative using the chain rule:

p'(t) = (10,000/((1+19e^(-t/5))^2)) * (-(19/5)*e^(-t/5))

Now we substitute t=4 into the derivative:

p'(4) = (10,000/((1+19e^(-4/5))^2)) * (-(19/5)*e^(-4/5))

Calculating the numerical value of p'(4), we get approximately -47.2. So, the fish population is changing at a rate of -47.2 fish per month at the end of 4 months.