A particular bullet train accelerates and decelerates at the rate of $4$ ft$/$s$2.$ Its maximum cruising speed is $60$ mi$/$h$.($Round your answers to three decimal places.$)$

$($a$)$What is the maximum distance $($in mi$)$ the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for $20$ minutes?

$($b$)$

Suppose that the train starts from rest and must come to a complete stop in $20$ minutes. What is the maximum distance $($in mi$)$

$($c$)$Find the minimum time $($in min$)$ that the train takes to travel between two consecutive stations that are $30$ miles apart.$)$ it can travel under these conditions?

$($d$)$The trip from one station to the next takes at minimum $37\mathrm{.}5$ minutes. How far $($in mi$)$ apart are the stations?