Answer
rising at about $\$ 1.627$ billion a year
Work Step by Step
If $t$ represents the number of years after 1999 and then the population in time $t$ can be written
\[
9200 t+961400=P(t)
\]
, the average annual income would be
\[
1400 t+30593=A(t)
\]
multiplying both together gives the total personal income at time $t$
\[
P(t) A(t)=T(t)
\]
The rate at which total personal income rises is $T^{\prime}(t)$
\[
\begin{aligned}
&P^{\prime}(t) A(t)+P(t) A^{\prime}(t)=T^{\prime}(t) \\
&=(9200 t+961400)^{\prime}(1400 t+30593)+(9200 t+961400)(1400 t+30593)^{\prime} \\
&=(1400 t+30593)9200+1400 (9200 t+961400)
\end{aligned}
\]
For $1999, t=0$ thus we can plug that in now , save a little work.
\[
9200(0+30593)+(0+961400) 1400=T^{\prime}(0)
\]
\[
=1,627,415,600
\]
It was rising at $\approx \$ 1.627$ billion a year.
The $P^{\prime}(t) A(t)$ part of the product rule is the part of the rate of change of total income that comes from the annual income from the additional population.
The $P(t) A^{\prime}(t)$ part of the product rule is the part of the rate of change of total income that comes from the increased income of the total population.
