Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

1. An open rectangular box is to be constructed from material that costs 3 dollars per ft$^2$ for the bottom and 1 dollar per ft$^2$ for its sides. Find the dimensions of the box of greatest volume that can be constructed for 36 dollars.