Canada Post accepts international parcels whose (Length+Girth) is less than or equal to 2 meters, and Length is less than or equal to 1 meter. Girth is defined as the perimeter of the cross section. We wish to ship a parcel of the shape of a triangular prism of length l meters. The cross section is a right triangle with catheti of lengths a and b meters. Assume the package walls are thin. What is the maximal volume of a parcel?

$\displaystyle leg_1=a=y, leg_2=b=z, length=x$

I provided a drawing via paint for you to envision my take on the problem

Hence,

$\displaystyle V(x, y, z)=\frac{1}{2}xyz$

Boundaries:

$\displaystyle (x+y+z+\sqrt{y^2+y^z})\leq{2}\\ x\leq{1}\\ x\geq{0}\\ y\geq{0}\\ z\geq{0}$

The critical points of the gradient of the volume equation comes up with (0, 0, 0), hence it is of no use. Okay, I have not had much experience with Lagrange multipliers, but here is my attempt.

$\displaystyle (x+y+z+\sqrt{z^2+y^2})={2}$

Hence, let

$\displaystyle L=0.5xyz+\lambda{(}x+y+z+\sqrt{y^2+z^2}-2)$

The critical points of L are determined via

$\displaystyle L_1=0.5yz+\lambda$

$\displaystyle L_2=0.5xz+\lambda{+}\frac{2y}{\sqrt{y^2+z^2}}$

$\displaystyle L_3=0.5xy+\lambda{+}\frac{2z}{\sqrt{y^2+z^2}}$

$\displaystyle L_4=x+y+z+\sqrt{x^2+y^2}-2$

At this point the system of equations look quite nasty, and moreoever I have been told that there is an easier route to solving this problem. Can someone suggest something?