Answer
$f(x)=\displaystyle \frac{900}{1+8(1.5)^{-x}}$
Work Step by Step
Logistic model: $f(x)=\displaystyle \frac{N}{1+Ab^{-x}}$,
where $N$ = limiting value.
While x is relatively small, $f$ behaves exponentially, $f(x)\displaystyle \approx(\frac{N}{1+A})b^{x}.$
We are given
$\left\{\begin{array}{l}
N=900,\\
b=100\%+50\%=1.50,\\
f(0)=100
\end{array}\right.$
We plug this in to obtain:
$f(0)=100\quad\Rightarrow\left\{\begin{array}{ll}
100 & =\dfrac{900}{1+Ab^{0}}\\
1+A & =\dfrac{900}{100}\\
A & =9-1=8
\end{array}\right.$
Thus we have:
$f(x)=\displaystyle \frac{900}{1+8(1.5)^{-x}}$
