Answer
$f(t)=\displaystyle \frac{10,000}{1+9999(1.40)^{-t}}$
Three weeks from now, $f(21)\approx 1049$ cases.
Work Step by Step
Logistic Model form: $\displaystyle \quad f(t)=\frac{N}{1+Ab^{-t}}$
$N$ is the limiting value, the total population. $N=10,000$
Now (t=0 days), we are given
$f(0)=1 \quad \Rightarrow \quad \left\{\begin{array}{ll}
1 & =\dfrac{10,000}{1+Ab^{-0}}\\
1+A & =10,000\\
A & =9999
\end{array}\right.$
The rate of increase was 40$\%$ per day, which corresponds to the factor:
$b=100\%+40\%=1.40$
$f(t)=\displaystyle \frac{10,000}{1+9999(1.40)^{-t}}$
Three weeks from now, $t=21 $ days, we have:
$f(21)\approx 1049$ cases.
