Answer
$\dfrac{8r^3}{3\sqrt 3}$
Work Step by Step
Need to apply Lagrange Multipliers Method to determine the maximum volume of a rectangular box.
we have $\nabla f=\lambda \nabla g$
The volume of a rectangular box is $V=abc$
The general equation for a rectangular box is $x^2+y^2+z^2=4r^2$
Re-write the above equations as: $z=\sqrt{4r^2-x^2-y^2}$
Since, $V=abc$ yields $V_x=0, V_y=0$
Also, $a=b=c=\dfrac{2r}{\sqrt 3}$
Therefore, the desired maximum volume of a rectangular box is
$V=abc=(\dfrac{2r}{\sqrt 3})^3$
$=(\dfrac{2r}{\sqrt 3})(\dfrac{2r}{\sqrt 3})(\dfrac{2r}{\sqrt 3})$
$=\dfrac{8r^3}{3\sqrt 3}$
