Problem 37 - Algebra and Trigonometry

Question

The logistic growth function $$ f(t)=\frac{100,000}{1+5000 e^{-t}} $$ describes the number of people, \(f(t),\) who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. 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?

Step-by-Step Solution

Answer

The initial impact of the epidemic led to 20 people being infected. To find the number of ill people after four weeks is a matter of solving the equation \(f(4) = \frac{100,000}{1 + 5000e^{-4}}\). The limiting size of the population that becomes infected, assuming sufficient time, is 100,000.

Step 1: Identify the initial number of patients when epidemic began

At the onset of the epidemic, t = 0 weeks. Substituting t = 0 into the logistic function \(f(t) = \frac{100,000}{1 + 5000e^{-t}}\), we find \(f(0) = \frac{100,000}{1 + 5000e^{-0}} = \frac{100,000}{1 + 5000} = 20\). So, when the epidemic started, there were 20 people infected.

Step 2: Identify the number of patients at the end of the fourth week

At the end of the fourth week, t = 4 weeks. Substituting t = 4 into the logistic function, we find \(f(4) = \frac{100,000}{1 + 5000e^{-4}}\). Solve this expression to get the number of infected people at week 4.

Step 3: Identify the limiting size of the population that becomes ill

The limiting size of the population that becomes ill is the maximum possible value of the function, which is the limit as t approaches infinity. In this case, we find that the limit of \(f(t) = \frac{100,000}{1 + 5000e^{-t}}\) as \(t → ∞\) is 100,000. This result means that theoretically, the number of infected people can't get above 100,000, no matter how much time passes.