$1$

A cardboard box for packing quantities of small foam balls is to be manufactured. The top, bottom, and front

faces must be of double weight $($i$.$e$.,$ two pieces of cardboard$).$ A problem posed is to find the dimensions

of such a box that maximize the volume for a given amount of cardboard, equal to $72$ sq$.$ ft$.$ Find the

optimal solution analytically using Lagrange mulitpliers. Next find the optimal solution using gradient

descent algorithm and verify your result using Matlab fmincon$()$ or Python scipy.optimize$().$