Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function $B$ models Ben's position on the track, measured in meters from the western end of the track, at time $t$, measured in seconds from the start of the ride. The table below gives values for $B(t)$ and Ben's velocity, $v(t)$, measured in meters per second, at selected times $t$.
| \begin{tabular}{ c } $t$ |
| (seconds) |
& 0 & 10 & 40 & 60 \hline
& 100 & 136 & 9 & 49 \hline
| $v(t)$ |
| (meters per second) |
& 2.0 & 2.3 & 2.5 & 4.6 \hline \end{tabular}
(a) Use the data in the table to approximate Ben's acceleration at time $t = 5$ seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of $\int_{0}^{60} |v(t)|\, dt$ in the context of this problem. Approximate $\int_{0}^{60} |v(t)|\, dt$ using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For $40 \leq t \leq 60$, must there be a time $t$ when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time $t$, the distance $L(t)$ between Ben and the light satisfies $(L(t))^2 = 12^2 + (B(t))^2$. At what rate is the distance between Ben and the light changing at time $t = 40$?