A high$-$speed bullet train accelerates and decelerates at the rate of $10$ft$/$s$2.$ Its maximum cruising speed is $90$mi$/$h$.($Round your answers to three decimal places.$)$

$($a$)$ What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for $15$ minutes?

mi$$

$($b$)$ Suppose that the train starts from rest and must come to a complete stop in $15$ minutes. What is the maximum distance it can travel under these conditions?

mi$$

$($c$)$ Find the minimum time that the train takes to travel between two consecutive stations that are $45$ miles apart.

min$$

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

mi$$