Attempt $1$:$5$ attempts remaining.

A two$-$day environmental clean up started at $9$ AM on the first day. The rate of change in the number of volunteers is shown in the following figure, where $\backslash (\backslash$boldsymbol$\{$t$\}\backslash )$ is the number of hours after $\backslash (\backslash$mathbf$\{9\}\backslash )$ AM on Day $1.$

$($You can click on a graph to enlarge it$.)$

a$.$ What are the units for the area of the region between the $\backslash ($ t $\backslash )-$axis and the graph of $\backslash ($ v$^\{\backslash$prime$\}($t$)\backslash )?$

volunteers per hour

volunteers

hours

b$.$ When was the number of volunteers increasing most rapidly?

At $1$ on the day.

c$.$ Write a sentence of interpretation for the point at $\backslash (\backslash$boldsymbol$\{$t$\}=\backslash$mathbf$\{28\}\backslash ).$

At $1$ on the day, the number of volunteers was by volunteers per hour.

d$.$ What does the area of the region between the graph of $\backslash ($ v$^\{\backslash$prime$\}($t$)\backslash )$ and the $\backslash (\backslash$boldsymbol$\{$t$\}\backslash )-$axis from $\backslash (\backslash$boldsymbol$\{$t$\}=\backslash$mathbf$\{0\}\backslash )$ to $\backslash (\backslash$boldsymbol$\{$t$\}=\backslash$mathbf$\{48\}\backslash )$ represent?

The area represents the in the number of volunteers over the two$-$day environmental clean up$.$