What is the maximal volume of a post package of length $L$, width $W$ and height $H$, subject to the following restrictions:
- $L+W+H \leq 90 $
- $L \leq 60$, $W \leq 60$, $H \leq 60$
Intuitively I would say $30^3$, but how do I find the solution mathematically?
Taking the partial derivatives: $\ V_L=WH=V_W=LH=V_H=LW=0 =>L=W=H$
And therefore the maximum volume is: $V=(90/3)^3=30^3$. However, here I have assumed that the maximum volume occurs for $L+W+H=90$,
How do I prove this?
I have also not incorporated the $L \leq 60$, $W \leq 60$, $H \leq 60$ restrictions.
Thanks in advance for any answers!