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The logistic growth function at right describes the number of​ people, ​f(t)​, who have become ill with influenza t weeks after its initial outbreak in a particular community. f (t )equals StartFraction 107 comma 000 Over 1 plus 4200 e Superscript negative t EndFraction.
a. How many people became ill with the flu when the epidemic​ began?
b. How many people were ill by the end of the fourth​ week?
c. What is the limiting size of the population that becomes​ ill?

Answer :

Answer:

Part (a): 25 people became ill with the flu when the epidemic​ began.

Part (b): 1373 people were ill by the end of the fourth​ week.

Part (c): The limiting size of the population that becomes​ ill is 107,000.

Step-by-step explanation:

Consider the provided logistic growth function

[tex]f (t )=\frac{ 107,000}{1 +4200e^{-t}}[/tex]

Part (a) How many people became ill with the flu when the epidemic​ began?

Substitute t = 0 in above function.

[tex]f (t )=\frac{ 107,000}{1 +4200e^{-0}}[/tex]

[tex]f (t )=\frac{ 107,000}{4201}[/tex]

[tex]f (t )=25.47[/tex]

Hence, 25 people became ill with the flu when the epidemic​ began.

Part (b) How many people were ill by the end of the fourth​ week?

Substitute t = 4 in above function.

[tex]f (t )=\frac{ 107,000}{1 +4200e^{-4}}[/tex]

[tex]f (t )=1373.10[/tex]

Hence, 1373 people were ill by the end of the fourth​ week.

Part (c) What is the limiting size of the population that becomes​ ill?

Substitute t = ∞ in above function.

[tex]f (t )=\frac{ 107,000}{1 +4200e^{-\infty}}[/tex]

[tex]f (t )=\frac{ 107,000}{1 +0}[/tex]

[tex]f (t )=107,000[/tex]

Hence, the limiting size of the population that becomes​ ill is 107,000.

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