Problem $2$

The function in Problem $1$ describes how many people are currently infected with a virus on any given day, which does not include people who were infected in the past and have since recovered or died. In this problem, we consider a function that describes the total number of cases since a pandemic began.

Suppose that the total number of cases $t$ days after a pandemic began, $C(t),$ can be approximated by a logistic function, leveling off at $8,000,000$ in the long run. Recall that a logistic function has the form

$C(t)=\frac{m}{1+(\frac{m}{c_0}-1)e^{-kmt}}$

where $m$ is the long$-$term maximum, $C_0$ is the initial amount, and $k$ is a constant. If the initial number of cases is $160$ and there have been $1,250$ total cases after $30$ days, find the following.

a$)$ Find a formula for $C(t)$ as a function of time. Show your work in finding the exact value of $k,$ and round any numbers in the final formula to four decimal places.