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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) = 200,000/1+2000e^-t
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?

Respuesta :

Answer:

Given function is:

[tex]f(t)=\frac{200000}{1+2000e^{-t}}[/tex]

Part A:

When t = 0

[tex]f(0)=\frac{200000}{1+2000e^{-0}}[/tex]

= [tex]\frac{200000}{1+2000e^{-0}}[/tex]

= [tex]\frac{200000}{1+2000}[/tex]

=  [tex]\frac{200000}{2001}[/tex]

= 99.95 rounding to 100 people.

Part B:

[tex]f(4)=\frac{200000}{1+2000e^{-4}}[/tex]

= [tex]\frac{200000}{1+2000(0.0183156)}[/tex]

= [tex]\frac{200000}{1+36.6312}[/tex]

= [tex]\frac{200000}{37.6312}[/tex]

= 5314.73

So, approximately 5,315 people were ill by the end of the 4th week.

Part C:

Let 't' be approaching infinity.

[tex]e^{-t}[/tex] will approach zero, so f(infinity)=[tex]\frac{200000}{1+0}[/tex]

f(infinity)=200000

Hence, the limit is that all of the persons can become ill.

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