The velocity of a particle, $P$, moving along the $x$-axis is given by the differentiable function $v_P$, where $v_P(t)$ is measured in meters per hour and $t$ is measured in hours. Selected values of $v_P(t)$ are shown in the table below. Particle $P$ is at the origin at time $t = 0$.
| \begin{tabular}{ c } $t$ |
| (hours) |
& 0 & 0.3 & 1.7 & 2.8 & 4 \hline
| $v_P(t)$ |
| (meters per hour) |
& 0 & 55 & -29 & 55 & 48 \hline \end{tabular}
(a) Justify why there must be at least one time $t$, for $0.3 \leq t \leq 2.8$, at which $v_P'(t)$, the acceleration of particle $P$, equals 0 meters per hour per hour.
(b) Use a trapezoidal sum with the three subintervals $[0, 0.3]$, $[0.3, 1.7]$, and $[1.7, 2.8]$ to approximate the value of $\int_0^{2.8} v_P(t)\, dt$.
(c) A second particle, $Q$, also moves along the $x$-axis so that its velocity for $0 \leq t \leq 4$ is given by $v_Q(t) = 45\sqrt{t}\cos\left(0.063t^2\right)$ meters per hour. Find the time interval during which the velocity of particle $Q$ is at least 60 meters per hour. Find the distance traveled by particle $Q$ during the interval when the velocity of particle $Q$ is at least 60 meters per hour.
(d) At time $t = 0$, particle $Q$ is at position $x = -90$. Using the result from part (b) and the function $v_Q$ from part (c), approximate the distance between particles $P$ and $Q$ at time $t = 2.8$.