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

Answer:

Given function is:

$f(t)=\frac{200000}{1+2000e^{-t}}$

Part A:

When t = 0

$f(0)=\frac{200000}{1+2000e^{-0}}$

= $\frac{200000}{1+2000e^{-0}}$

= $\frac{200000}{1+2000}$

=  $\frac{200000}{2001}$

= 99.95 rounding to 100 people.

Part B:

$f(4)=\frac{200000}{1+2000e^{-4}}$

= $\frac{200000}{1+2000(0.0183156)}$

= $\frac{200000}{1+36.6312}$

= $\frac{200000}{37.6312}$

= 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)=$\frac{200000}{1+0}$

f(infinity)=200000

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