The number of people in a community who became infected during an epidemic $\mathrm{t}$ weeks after its outbreak is given by the function $\mathrm{f}(\mathrm{t})=\frac{30,000}{1+\mathrm{a} e^{-\mathrm{kt}}}$, 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. 12964 (Simplify your answer. Do not round until the final answer. Then round to the nearest whole number as needed.) b. After how many weeks will 9,466 people be infected? weeks (Simplify your answer. Do not round until the final answer. Then round to the nearest whole number as needed.)

Step 1 ：Given the function \(f(t)=\frac{30000}{1+ae^{-kt}}\), where \(f(t)\) is the number of people infected after \(t\) weeks, \(a\) and \(k\) are constants, and \(e\) is the base of the natural logarithm.

Step 2 ：We know that 5000 people were infected initially, so when \(t=0\), \(f(t)=5000\). Substituting these values into the function gives us \(5000=\frac{30000}{1+a}\). Solving for \(a\) gives \(a=5\).

Step 3 ：We also know that 8419 people had been infected by the end of the fourth week, so when \(t=4\), \(f(t)=8419\). Substituting these values into the function gives us \(8419=\frac{30000}{1+5e^{-4k}}\). Solving for \(k\) gives \(k\approx0.2417\).

Step 4 ：a. To find the number of people infected after 8 weeks, we substitute \(t=8\), \(a=5\), and \(k\approx0.2417\) into the function, which gives us \(f(8)\approx12964\). So, approximately 12964 people will be infected after 8 weeks.

Step 5 ：b. To find out after how many weeks will 9466 people be infected, we set \(f(t)=9466\), \(a=5\), and \(k\approx0.2417\), and solve for \(t\). This gives us \(t\approx6\). So, approximately 6 weeks will pass before 9466 people are infected.