The number of people in a community who became infected during an epidemic t weeks after its outbreak is given by the function $f(t)=\frac{30,000}{1+a e^{-k t}}$, where 30,000 people of the community are susceptible to the disease. Assume that 5000 people were infected initially and 8419 had been infected by the end of the fourth week. Complete parts a. and b. a. Find the number of people infected after 8 weeks. (Simplify your answer. Do not round until the final answer. Then round to the nearest whole number as needed.)

Step 1 ：We are given that 5000 people were infected initially and 8419 had been infected by the end of the fourth week. We can use these two conditions to solve for the constants a and k in the function \(f(t)=\frac{30,000}{1+a e^{-k t}}\).

Step 2 ：First, we know that at t=0 (the start of the outbreak), the number of infected people was 5000. So we can set up the equation \(5000 = \frac{30,000}{1+a}\) to solve for a. From this equation, we find that a = 5.

Step 3 ：Second, we know that at t=4 (four weeks after the outbreak), the number of infected people was 8419. So we can set up the equation \(8419 = \frac{30,000}{1+5 e^{-4k}}\) to solve for k. From this equation, we find that k = \(\log(21581^{3/4}*42095^{1/4}/21581) - i\pi/2\).

Step 4 ：After we find the values of a and k, we can substitute them into the function and find the number of people infected after 8 weeks by evaluating \(f(8)\).

Step 5 ：Substituting the values of a and k into the function, we get \(f = \frac{1771989025000}{136689561}\).

Step 6 ：Evaluating this function at t=8, we find that the number of people infected after 8 weeks is approximately 12963.6016974259.

Step 7 ：Rounding to the nearest whole number, we find that the number of people infected after 8 weeks is approximately \(\boxed{12964}\).