Answer
14.5 inches
Work Step by Step
We will build a piece-wise function $f(t)$ to model the situation.
For the first two hours the slope is 1, so the function would be:
$f(t)=t$, where $0\leq t\leq 2$
For the next six hours the slope is 2. Determine the equation using the point $(2,2)$ and the slope 2:
$y-2=2(t-2)$
$y-2=2t-4$
$y=2t-4+2$
$y=2t-2$
So the function would be:
$f(t)=2t-2$, where $2\lt t\leq 8$
For the last hour the slope is 0.5. Determine the equation using the point $(8,2(8)-2)=(8,14)$ and the slope 0.5:
$y-14=0.5(t-8)$
$y-14=0.5t-4$
$y=0.5t-4+14$
$y=0.5t+10$
So the function would be:
$f(t)=0.5t+10$, where $8\lt t\leq 9$
The function is:
$f(t)=\begin{cases}
t,0\leq t\leq 2\\
2t-2,2\lt t\leq 8\\
0.5t+10,8\lt t\leq 9
\end{cases}$
Graph the function.
The total accumulation is:
$f(9)=0.5(9)+10=4.5+10=14.5$
