Question

Question 9 B0/1 pt 99 97 Details The function f(t) = 575,000 1 + 4000e- describes the number of people, f(t), who have become ill with ebola t weeks after the initial outbreak in a particular community. How many people became ill with ebola when the epidemic began? Round to the nearest whole number of people. 15 urces How many people were infected 6 weeks after the initial breakout? Round to the nearest whole number of people. What is the limiting size of the infected population? That is, how many people were infected at t - ? Round to the nearest whole number of people. Question Help: Video Submit Question

Answer 1

Final Answers:

1. 575,000 people became ill with Ebola when the epidemic began.

2. Approximately 601,473 people were infected 6 weeks after the initial outbreak.

3. The limiting size of the infected population was 575,000 people.

Step-by-step explanation:

The function
`\(f(t) = 575,000 * (1 + 4000e^(-t))\)`models the number of people, \(f(t)\), who contracted Ebola
`\(t\)` weeks after the initial outbreak. To determine the number of individuals infected when the epidemic commenced (t = 0), substitute \(t = 0\) into the function. Doing so yields
`\(f(0) = 575,000 * (1 + 4000e^(0)) = 575,000 * (1 + 4000) = 575,000 * 4001 = 2,301,500,000\).`Rounding to the nearest whole number, approximately 575,000 people were initially affected by Ebola.

To ascertain the count of infections at 6 weeks post-outbreak, evaluate \
`(f(6)\)`. Plugging in \(t = 6\) into the function gives
`\(f(6) = 575,000 * (1 + 4000e^(-6)) \approx 601,472.92\).`Rounded to the nearest whole number, roughly 601,473 individuals contracted Ebola after 6 weeks.

As
`\(t\)`approaches infinity, the function's exponential term
`\(e^(-t)\)`diminishes toward zero, leaving only the constant term of 575,000. Thus, the limiting size of the infected population is the initial value of 575,000 individuals. This represents the maximum capacity of infected individuals in the community as time progresses indefinitely.

Therefore, at the onset, 575,000 individuals were affected, with the count reaching approximately 601,473 after 6 weeks. Eventually, the infected population stabilizes at the initial figure of 575,000 as the epidemic progresses towards its limit.""

Answer 2

Approximately 2,300,575,000 people became ill with Ebola when the epidemic began. Approximately 6,285,625 people were infected 6 weeks after the initial outbreak. The limiting size of the infected population is 575,000 people.

The number of people who became ill with Ebola when the epidemic began can be found by substituting t = 0 t=0 into the given function
`f(t) = 575,000(1 + 4000e^(-t) ).` Since
`e^0 = 1,`

the expression simplifies to:

f(0) = 575,000(1 + 4000)

Now, calculate this:

f(0) = 575,000 * 4001

f(0) = 2,300,575,000

So, approximately 2,300,575,000 people became ill with Ebola when the epidemic began.

To find the number of people who were infected 6 weeks after the initial outbreak (t = 6), substitute t = 6 into the function:

f(6) = 575,000(1 + 4000e^(-6))

Calculate this expression:

f(6) ≈ 575,000 * (1 + 4000 * 0.00247875)

f(6) ≈ 575,000 * (1 + 9.915)

f(6) ≈ 575,000 * 10.915

f(6) ≈ 6,285,625

So, approximately 6,285,625 people were infected 6 weeks after the initial outbreak.

The limiting size of the infected population as t approaches infinity (t -> ∞) is:

`\lim_(n \to \infty) f(t)`= 575,000 * (1 + 4000 * 0) = 575,000

So, the limiting size of the infected population is 575,000 people.