Re: Geometric Optimization

Since the base is square, the two sides are equal: let "b" represent the length of a side of the base, in feet. Let "h" represent the height of the box, in feet. Then we have a square base, with area $\displaystyle b^2$ square feet and four sides with area bh square feet. Since "the base will cost 7 dollars a square foot and the sides will cost 9 dollars a square foot" the base will cost $\displaystyle 7b^2$ dollars and each of the four sides will cost $\displaystyle 9bh$ dollars so the total cost is $\displaystyle 7b^2+ 36bh$ dollars. That is the function you want to minimize.

Since you did not show any attempt yourself to do this problem, or tell us what methods you have learned to find "minimum" values of a function, I don't know how to tell you to continue.