Answer
The average juice volume change in the glass in 5 seconds equals the average value of the juice flow rate in the glass $=28\mathrm{ml} $.
Work Step by Step
From the figure, we will find function of the change of the volume with time:
And then we will find the formula of the volume by integrating the function.
When $0 \leq t \leq 1 V(t)=20 t$
When $1 \leq t \leq 3 \quad V(t)=-20+40 t$
When $3 \leq t \leq 5 V(t)=-110+100 t-19 t^{2}$
Therefore, the average rate of change of juice volume in glass during these five is:
\[
V(t)=\left\{\begin{array}{cc}
-20+40 t & 1 \leq t \leq 3 \\
-110+100 t -10 t^{2}& 3 \leq t \leq 5 \\
20 t^{2} & 0 \leq t \leq 1
\end{array}\right.
\]
The average value of the juice-to-glass flow rate is:
\[
\frac{d V}{d t}=\frac{-V(0)+V(5)}{-0+5}=\frac{-0+140}{5}=28 \mathrm{ml}
\]
\[
\begin{aligned}
f_{a v g} &=\frac{1}{-a+b} \int_{a}^{b} f(t) d t \\
&=\frac{1}{5-0}\left(40 \int_{0}^{1} t d t+40 \int_{1}^{3} d t-20 \int_{3}^{5}(-5+t) d t\right) \\
&=\frac{1}{5}\left(\left.40 \frac{t^{2}}{2}\right|_{0} ^{1}+\left.40 t\right|_{1} ^{3}-\left.20\left(\frac{-5 t+t^{2}}{2}\right)\right|_{3} ^{5}\right)
\end{aligned}
\]
\[
\begin{array}{l}
=\frac{1}{5}(40+80+20) \\
=28
\end{array}
\]
