Answer
$M(t)=0.8t^{4}-6t^{2}+10t+1$
Work Step by Step
$M(t) $ is an antiderivative of the given instaneous rate of change $m(t).$
$M(t)=\displaystyle \int(3.2t^{3}-12t+10)dt=3.2\cdot\frac{t^{4}}{4}-12\cdot\frac{t^{2}}{2}+10t+C$
$M(t)=0.8t^{4}-6t^{2}+10t+C,$
which is a collection of functions. To find the exact function, we find $C.$
The start of the year 2005 is $t=0$ years after 2005. We are given
$M(0)=1\qquad $million members.
$1=0.8($0$)-6(0)+10(0)+C$
$C=1$
Thus,
$M(t)=0.8t^{4}-6t^{2}+10t+1$
