Answer
$\dfrac{8r^3}{3\sqrt 3}$
Work Step by Step
Volume of a rectangular box is given by $V=abc$
which has general equation: $x^2+y^2+z^2=4r^2$
This implies $z=\sqrt{4r^2-x^2-y^2}$
This gives $V_x=0, V_y=0$
and
$a=b=c=\dfrac{2r}{\sqrt 3}$
Hence, the maximum volume of a rectangular box will be
$V=abc=(\dfrac{2r}{\sqrt 3})^3=\dfrac{8r^3}{3\sqrt 3}$
