Determine the carrying capacity of the environment.

Question

The logistic growth model   represents the population (in grams) of a bacterium after t hours.

(a) Determine the carrying capacity of the environment.
(b) What is the growth rate of the bacteria?
(c) Determine the initial population size.
(d) What is the population after 9 hours?
(e) When will the population be 700 grams?
(f) How long does it take for the population to reach one-half the carrying capacity?

Answer 1

Step 1 :

(a)

The logistic growth model of bacterium after hours is grams.

The carrying  capacity of the environment can be find out by substituting in logistic growth model.

The carrying  capacity of the environment is grams.

Solution :

The carrying  capacity of the environment is grams.

Answer 2

Step 1 :

(b)

The standard logistic growth model of population after hours is .

The logistic growth model of bacterium after hours is grams.

Compare the logistic model with standard logistic model and .

The growth rate is for standard logistic model is .

Therefore the growth rate of the bacteria is per hour.

Solution :

The growth rate of the bacteria is per hour.

Answer 3

Step 1 :

(c)

The logistic growth model of bacterium after hours is grams.

The initial population size can be find out by substituting in logistic growth model.

Solution :

The initial population size of bacteria is 30 grams.

Answer 4

Step 1 :

(d)

The logistic growth model of bacterium after hours is grams.

The population size after hours can be find out by substituting in logistic growth model.

Solution :

The population size after hours is grams.

Answer 5

Step 1 :

(e)

The logistic growth model of bacterium after hours is grams.

The time when population size reaches grams can be find out by substituting in logistic growth model.

Solution :

The population size of bacteria is   grams after hours.

Answer 6

Step 1 :

(f)

The logistic growth model of bacterium after hours is grams.

The carrying  capacity of the environment is grams.

One-half the carrying capacity means grams.

The time when population size reaches grams can be find out by substituting in logistic growth model.

Solution :

The population size of bacteria reaches one-half the carrying capacity after hours.