Question

The population of a culture of bacteria is modeled by the logistic equation
.
To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity? What is the carrying capacity? What is the initial population for the model? Why a model like , where is the initial population, would not be plausible? What are the virtues of the logistic model?
Go to www.desmos.com/calculator and type
y = 14250 / (1 + 29 . e-0.62 x). {0 < x < 15} {0 < y < 15000}
y = 14300 {0 < x < 15}
(you will find the command “ ” in the desmos calculator after selecting “14250”, or you type “/” after selecting “14250”, and you will also find the function “exp” ). Adjust the x and y axes settings to 0 < x < 15 and 0 < y < 15000. Plot the graph you have obtained (you can use a screenshot, save as image, and copy it into word). If you need, or if you want, go to the Course Forum and tell us something about this plotting task.

Answer 1

The carrying capacity:

$y_{max}=14250$ when $x\to\infin$

The initial population:

$y_0=475$ when $x=0$

75% of its carrying capacity: $0.75\cdot14250=10687.5$

$10687.5=\frac{14250}{1+29e^{-0.62x}}$

$x=-ln((\frac{14250}{10687.5}-1)/29)/0.62=7.2$ days

Why a model like , where is the initial population, would not be plausible?

Because in this case number of days=0, that is, the process has not started yet.

The virtues of the logistic model: simplicity, flexibility.