Answer
a) $\$24.65$ millions
b) $\$54,222.156$ millions
c) $N(t)= \frac{55.125 t^2-6435.607 t+186914.286}{0.464 t-12.47}$
d) $\$1008.75$
e) $1940$ and $1997$
Work Step by Step
Given \begin{equation}
\begin{aligned}
P(t)&=0.464 t-12.47\\
S(t)&=55.125 t^2-6435.607 t+186914.286
\end{aligned}
\end{equation} where $P(t)$ is the population of the of the state of California in millions $t$ years since $1900$, and $S(t)$ is the amount California spent in millions of dollars$t$ years since $1900$.
a) Find $P(t)$ when $t= 80$ for the year $1980$ since $1900$. \begin{equation}
\begin{aligned}
P(90)&=0.464 \cdot 80-12.47\\
&= 24.65.
\end{aligned}
\end{equation} The population of the state of California was about $\$24.65$ millions.
b) Find $S(t)$ when $t= 90$ for the year $1990$ since $1900$. \begin{equation}
\begin{aligned}
S(90)&=55.125 \cdot 90^2-6435.607 \cdot 90+186914.286\\
&= 54222.156.
\end{aligned}
\end{equation} The amount that the state of California spent in $1990$ was about $\$54222.156$ millions.
c) Let $N(t)$ represents the spending per capital per person of the state of California $t$ years since $1900$. Then, we have $N(t) = \frac{S(t)}{P(t)}$
\begin{equation}
\begin{aligned}
N(t)&= \frac{55.125 t^2-6435.607 t+186914.286}{0.464 t-12.47}.
\end{aligned}
\end{equation} d) The per capital spending in $1980$ is given by: \begin{equation}
\begin{aligned}
N(80)&= \frac{55.125 \cdot 80^2-6435.607 \cdot 80+186914.286}{0.464 \cdot 80-12.47}\\
&= 1008.75.
\end{aligned}
\end{equation} The per capita spending in California in $1980$ was about $\$1008.75$.
e) Set $N(t) = 2500$ and solve for $t$. Alternatively, graph both $N(t)$ and $g(t) = 2500$ in the same window and estimate $t$. \begin{equation}
\begin{aligned}
N(t)&= 2500\\
\frac{55.125 t^2-6435.607 t+186914.286}{0.464 t-12.47}&= 2500\\
55.125 t^2-6435.607 t+186914.286&=2500(0.464 t-12.47)\\
55.125 t^2-6435.607 t+186914.286-2500(0.464 t-12.47)&= 0\\
55.125 t^2-6435.607 t+186914.286-1160 t+31175&=0\\
55.125 t^2-7595.607 t+218089.286&= 0\\
\frac{55.125 t^2-7595.607 t+218089.286}{55.125 }&= 0\\
t^2-137.78879t+3956.268227&=0
\end{aligned}
\end{equation} \begin{equation}
\begin{aligned}
t&=\frac{-(-137.78879) \pm \sqrt{(-137.78879)^2-4 \cdot 1 \cdot 3956.268227}}{2 \cdot 1}\\
&= \frac{137.78879 \pm \sqrt{3160.67774 }}{2 }.
\end{aligned}
\end{equation} This gives $t=97.00434 \ldots, t=40.78444$.
These correspond to the year $1941$ and $1997$.The per capita spending in California reached $\$2500$ in about $1940$ and again in about $1997$.
