Answer
Maximum value : $3534 cm^{3}$, Minimum value: $2948 cm^{3}$
Work Step by Step
We need to apply the Lagrange Multipliers Method to determine the maximum and minimum values of a rectangular box of maximum volume.
we have $\nabla f=\lambda \nabla g$
The volume of a box is $V=xyz$
Consider $f(x,y,z)=V=xyz$
Surface area, $S=2xy+2yz+2zx$
From the given question, $S=1500 cm^2$
Now, $\nabla f=\lt yz,xz,xy \gt$ and $\lambda \nabla g=\lambda \lt 2x,2y,2z \gt$
Total length of the edge, $4x+4y+4z=200$
Simplify to get the value of $\lambda =x=y$
Therefore, the volume of a box is $V=xyz$; when $x=\dfrac{5(10+\sqrt {10})}{3}$
Hence, Maximum value : $3534 cm^{3}$, Minimum value: $2948 cm^{3}$
