A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function $f$, where $f(t)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. Selected values of $f(t)$ are given in the table.

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
$t$ \\
(seconds) \\
\end{tabular} & 0 & 60 & 90 & 120 & 135 & 150 \\
\hline
\begin{tabular}{ c }
$f(t)$ \\
(gallons per second) \\
\end{tabular} & 0 & 0.1 & 0.15 & 0.1 & 0.05 & 0 \\
\hline
\end{tabular}
\end{center}

(a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\, dt$ in the context of the problem. Use a right Riemann sum with the three subintervals $[60,90]$, $[90,120]$, and $[120,135]$ to approximate the value of $\int_{60}^{135} f(t)\, dt$.

(b) Must there exist a value of $c$, for $60 < c < 120$, such that $f'(c) = 0$? Justify your answer.

(c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t) = \left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^{2}\right)$ for $0 \leq t \leq 150$. Using this model, find the average rate of flow of gasoline over the time interval $0 \leq t \leq 150$. Show the setup for your calculations.

(d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem.