Answer
$St)=-15t^{3}+450t^{2}+4200t,$
$54,720 $ million gallons sold
Work Step by Step
The instaneous rate of change of $S(t)$ is
$S'(t)=s(t)=-45t^{2}+900t+4,200$ million gallons per year
$S(t)=\displaystyle \int(-45t^{2}+900t+4,200)dt$
$=-45\displaystyle \cdot\frac{t^{3}}{3}+900\cdot\frac{t^{2}}{2}+4,200t+C$
$=-15t^{3}+450t^{2}+4200t+C$
To find C, use the given hint:
$S(0)=0$
$0=0+0+0+C$
$C=0$
$St)=-15t^{3}+450t^{2}+4200t$
The start of the year 2008 is $t=8$ years after the start of 2000.
$S(7)=-15(8^{3})+450(8^{2})+4200(8)=54,720$
