Answer
$S(t)=-0.06t^{3}+4.5t^{2}+15t,$
$540$ gallons per capita sold
Work Step by Step
The instantaneous rate of change of $S(t)$ is
$S'(t)=s(t)=-0.18t^{2}+3t+15$ gallons per year
$S(t)=\displaystyle \int(-0.18t^{2}+3t+15)dt$
$=-0.18\displaystyle \cdot\frac{t^{3}}{3}+3\cdot\frac{t^{2}}{2}+15t+C$
$=-0.06t^{3}+1.5t^{2}+15t+C$
To find C, use the given hint:
$S(0)=0$
$0=0+0+0+C$
$C=0$
$S(t)=-0.06t^{3}+1.5t^{2}+15t$
The end of the year 2009 is $t=10$ years after the start of 2000.
$S(7)=-0.06(10^{3})+1.5(10^{2})+15(10)=540$ gallons per capita
