The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.
| \begin{tabular}{ c } $t$ |
| (hours) |
&
| $R ( t )$ |
| (gallons per hour) |
\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int _ { 0 } ^ { 24 } R ( t ) d t$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t , 0 < t < 24$, such that $R ^ { \prime } ( t ) = 0$ ? Justify your answer.
(c) The rate of water flow $R ( t )$ can be approximated by $Q ( t ) = \frac { 1 } { 79 } \left( 768 + 23 t - t ^ { 2 } \right)$. Use $Q ( t )$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.