From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A ( t ) = 450 \sqrt { \sin ( 0.62 t ) }$, 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 $N ( t ) = \int _ { a } ^ { t } ( A ( x ) - 400 ) d x$, 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.