A carpenter has been asked to build an open box with a square base, where an open box means a box without a top. The sides of the box will cost $6 per square meter, and the base will cost $3 per square meter. What are the dimensions of the box of maximal volume that can be constructed for $144 ?
If x is the length of one of the sides of the base and z is the height of the box. Find a function f(x) for the volume of the box.
NOTE: The function should just be in terms of x.
Let x∗ and z∗ be the dimensions that maximize the volume of the box.