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. <IMAGE>


{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.


a. Graph P using a graphing utility. Experiment with different windows until you produce an S-shaped curve characteristic of the logistic model. What window works well for this function? 

Verified step by step guidance

1. Identify the parameters in the logistic growth function P(t) = 400,000 / (50 + 7950e^(-0.5t)), where P₀ = 50, K = 8000, and r₀ = 0.5.
2. Set up a graphing utility, such as Desmos or a graphing calculator, to input the logistic function P(t).
3. Choose an appropriate window for the graph. A good starting point for the x-axis (time t) could be from 0 to 20 years, and for the y-axis (population P), from 0 to 8000 fish, since this is the carrying capacity.
4. Plot the function and observe the shape of the curve. Adjust the window settings if necessary to better visualize the S-shaped curve characteristic of logistic growth.
5. Experiment with different values for the x-axis and y-axis limits to find the best view of the curve, ensuring that the population approaches the carrying capacity as time increases.