If [tex]t[/tex] and [tex]r[/tex] are inverse proportional to one another, then for some constant volume [tex]V[/tex] we have
[tex]V = tr[/tex]
It takes 45 min to empty a tank containing at 300 gal/min, so the tank contains
[tex]V = (45\,\mathrm{min}) \left(300\dfrac{\rm gal}{\rm min}\right) = 13500\,\mathrm{gal}[/tex]
of liquid.
If it's emptied at 500 gal/min, it would take
[tex]13500\,\mathrm{gal} = t \left(500\dfrac{\rm gal}{\rm min}\right) \implies t = \boxed{27\,\mathrm{min}}[/tex]
