Answer to Question #151077 in Algebra for isaac

Question

The population of a culture of bacteria is modeled by the logistic equation

P(t)= \frac{14,250}{1+29e^{-0.62t}.

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 P(t)=P_0 \ e^{Kt} , where P_0 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 “\div” 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

• To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity?

$P(t)= \frac{14,250}{1+29e^{-0.62t}}.$

75% = 3 / 4. By what number do you divide 14250 so that you get 3/4 of it? By 4/3. So,

$1+29e^{-0.62t} = \frac{4}{3}$

$29e^{-0.62t} = \frac{1}{3}$

$e^{-0.62t} = \frac{1}{87}$

ln(1/87) = -0.62t Rounded to the nearest tenth t=ln(1/87)/(-0.62)=7.2.

• What is the carrying capacity?

the carrying capacity is the upper bound of the logistic function. If you express a logistic function as $\frac{c}{1 + b^x}$ the carrying capacity is c.

In this problem, the carrying capacity is 14,250

• What is the initial population for the model?

It is the population at time zero, when t=0:

$14250/(1+29*e^0)=14250/(1+29*1)=475$

• Why a model like $P(t)=P_0 \ e^{Kt} ,$ where $P_0$ is the initial population, would not be plausible? What are the virtues of the logistic model?

The answer to this question should really be tuned in to your classroom discussions, because many explanations are possible. Basically, the difference between $P(t)=P_0 \ e^{Kt} ,$ and the logistic function is that the first is unlimited exponential growth and the second is growth in the realistic conditions of limited resources. One virtue of the logistic model is that it is realistic, because we live in the reality of limited resources.