Answer
Maximum value : $3533.34$ $cm^{3}$, Minimum value: $2947.94$ $cm^{3}$
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, $S=1500 cm^2$
Surface area, $S=2xy+2yz+2zx=1500 cm^2$ or,
$xy+yz+zx=750 cm^2$
Now, $\nabla f=\lt yz,xz,xy \gt$ and $\lambda \nabla g=\lambda \lt 2x,2y,2z \gt$
Total length of the edge can be calculated as: $4x+4y+4z=200$ or, $x+y+z=50$
After solving equations $x+y+z=50$ and $xy+yz+zx=750 cm^2$, we get $z=50-x-y$
Simplify to get the value of $\lambda =x=y$. Therefore, the volume of a box is $V=xyz$; when $x=y=\dfrac{5(10+\sqrt {10})}{-3}$ or, $x=y=\dfrac{-50 \pm 5\sqrt {10}}{-3}$
The critical points of $V(x,y)$ are $(\dfrac{-50+5\sqrt {10})}{-3},\dfrac{-50+5\sqrt {10})}{-3})$ and $(\dfrac{-50-5\sqrt {10})}{-3},\dfrac{-50-5\sqrt {10})}{-3})$
Also,
$V(\dfrac{-50+5\sqrt {10})}{-3},\dfrac{-50+5\sqrt {10})}{-3})=3533.34$ $cm^{3}$ and $(\dfrac{-50-5\sqrt {10})}{-3},\dfrac{-50-5\sqrt {10})}{-3})=2947.94$ $cm^{3}$
Hence, Maximum value : $3533.34$ $cm^{3}$, Minimum value: $2947.94$ $cm^{3}$
