I can't solve these questions, help me out.

Question

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

P(t) = (14,250) / (1 + 29 e^{– 0.62t} )  

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

b. What is the carrying capacity?

c. What is the initial population for the model?

d. Why a model like P(t) = P₀ e^Kt, where P₀ is the initial population, would not be plausible?

d. What are the virtues of the logistic model?

Go to desmos/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).

(Please help me with the graph)

Answer 1

Read more here:

https://www.ck12.org/book/CK-12-Precalculus-Concepts/section/3.7/

I'll start with b: the carrying capacity is the upper bound of the logistic function. If you express a logistic function as c/(1+a*b^x), the carrying capacity is c.

In this problem, the carrying capacity is 14,250.

a. 75% is 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))=4/3.

29e^(–0.62t)=4/3-1

e^(–0.62t)=(1/3)/29=1/87

ln(1/87)=-0.62t

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

c. What is the initial population? It is the population at time zero, when t=0:

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

d-c. 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) = P0*eKt 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.

For the graph, follow the instructions they provided, and it will work. If you use t for the variable, Desmos will ask you to create a slider. Otherwise, you can use x in place of t.