## Calculus 8th Edition

$\dfrac{8r^3}{3\sqrt 3}$
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 $V=abc=(\dfrac{2r}{\sqrt 3})^3$ $=(\dfrac{2r}{\sqrt 3})(\dfrac{2r}{\sqrt 3})(\dfrac{2r}{\sqrt 3})$ $=\dfrac{8r^3}{3\sqrt 3}$