The function $N(t)=\frac{30,000}{1+40 e^{-2.5 t}}$ describes the number of people, $N(t)$, who become ill with a virus $t$ weeks after its initial outbreak in a town with 30,000 inhabitants. The horizontal asymptote in the graph indicates that there is a limit to the epidemic's growth. Complete parts (a) through (c) below. a. How many people became ill with the virus when the epidemic began? (When the epidemic began, $t=0$.) When the epidemic began, approximately people were ill with the virus. (Round to the nearest person as needed.) b. How many people were ill by the end of the fourth week? By the end of the fourth week, approximately people were ill with the virus. (Round to the nearest person as needed.) c. Why can't the spread of an epidemic simply grow indefinitely? A. The number of ill people cannot exceed the population of the town. B. A treatment for the virus is eventually found. C. The people of the town gradually develop an immunity to the virus. D. All the ill people die from the virus. E. All of the above.

Step 1 ：Given the function \(N(t)=\frac{30,000}{1+40 e^{-2.5 t}}\), which describes the number of people, \(N(t)\), who become ill with a virus \(t\) weeks after its initial outbreak in a town with 30,000 inhabitants.

Step 2 ：For part a, we need to substitute \(t=0\) into the function to find the number of people who were ill when the epidemic began. This gives us \(N(0)=\frac{30,000}{1+40 e^{-2.5 \times 0}}\), which simplifies to approximately 732 people.

Step 3 ：For part b, we need to substitute \(t=4\) into the function to find the number of people who were ill by the end of the fourth week. This gives us \(N(4)=\frac{30,000}{1+40 e^{-2.5 \times 4}}\), which simplifies to approximately 29946 people.

Step 4 ：For part c, the question is asking why the spread of an epidemic can't grow indefinitely. This is a theoretical question and doesn't require any calculations. The answer is that the number of ill people cannot exceed the population of the town. This is because the function \(N(t)\) has a horizontal asymptote at \(N=30000\), which represents the total population of the town. The number of ill people can't exceed this number.

Step 5 ：Final Answer: a. When the epidemic began, approximately \(\boxed{732}\) people were ill with the virus. b. By the end of the fourth week, approximately \(\boxed{29946}\) people were ill with the virus. c. The spread of an epidemic can't grow indefinitely because \(\boxed{\text{The number of ill people cannot exceed the population of the town.}}\)