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?
I provided a drawing via paint for you to envision my take on the problem
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.
The critical points of L are determined via
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?