From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A(t) = 450\sqrt{\sin(0.62t)}$, where $t$ is the number of hours after 5 A.M. and $A(t)$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.
(a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. $(t=1)$ to 10 A.M. $(t=5)$.
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ($t=1$) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever $A(t) \geq 400$. The number of vehicles in line at time $t$, for $a \leq t \leq 4$, is given by $N(t) = \int_{a}^{t}(A(x) - 400)\,dx$, where $a$ is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval $a \leq t \leq 4$. Justify your answer.