subtract (1) from (3)
$c^2 - a^2 = 20$
add this to (2)
$2c^2 = 45$
$c = \sqrt{\dfrac{45}{2}} = 3\sqrt{\dfrac 5 2}$
plug this into (2)
$a^2 + \dfrac{45}{2} = 25$
$a^2 = \dfrac{5}{2}$
$a = \sqrt{\dfrac{5}{2}}$
plug this into (1)
$\dfrac 5 2 + b^2 = 16$
$b^2 = \dfrac{27}{2}$
$b = \sqrt{\dfrac{27}{2}}=3\sqrt{\dfrac 3 2}$
$V = 8abc = 8\left(\sqrt{\dfrac{5}{2}}\right)\left(3\sqrt{ \dfrac 3 2}\right)\left(3\sqrt{ \dfrac 5 2}\right)=90\sqrt{6}$