Question

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?

Answer 1

Given function is:

`f(t)=(200000)/(1+2000e^(-t))`

Part A:

When t = 0

`f(0)=(200000)/(1+2000e^(-0))`

=
`(200000)/(1+2000e^(-0))`

=
`(200000)/(1+2000)`

=
`(200000)/(2001)`

= 99.95 rounding to 100 people.

Part B:

`f(4)=(200000)/(1+2000e^(-4))`

=
`(200000)/(1+2000(0.0183156))`

=
`(200000)/(1+36.6312)`

=
`(200000)/(37.6312)`

= 5314.73

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

Part C:

Let 't' be approaching infinity.

`e^(-t)` will approach zero, so f(infinity)=
`(200000)/(1+0)`

f(infinity)=200000

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