Answer
a) $p(x)=1154.16(1.20112)^x $
b) $\$ 1154.16$
c) $20.112\%$
Work Step by Step
a) Let $P=f(x)=a(b)^x $ for the increasing exponential function as shown in the figure. The values of $a$ and $b$ can be found from the points, $(3,2000)$ and $(8, 5000)$ . We make use of the ratio of $
f(3)=a(b)^3=2000$, and $f(8)=a(b)^8=5000$
$$
\begin{aligned}
\frac{f(8)}{f(3)} & =\frac{a b^8}{a b^3}=\frac{5000}{2000} \\
b^5 & =\frac{5}{2}=2.5 \\
\left(b^5\right)^{1 / 5} & =(2.5)^{1 / 5} \\
b &\approx1.20112 .
\end{aligned}
$$ and $$
\begin{aligned}
a\left((2.5)^{1 / 5}\right)^{3} & =2000 \\
a & =\frac{2000}{(1.20112)^3} \\
& =1154.160 .
\end{aligned}
$$ Hence,
$p(x)=1154.16(1.20112)^x $
b) The initial balance is $\$ 1154.16$.
c) $$
\begin{aligned}
&b=1+r .\\
&\begin{aligned}
& 1.20112=1+r \\
& 0.20112=r\\
& 20.112\%=r
\end{aligned}
\end{aligned}
$$
The annual interest rate is $20.112\%$.
