A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing 1 dollar/sq ft, and the top is to be constructed of glass costing 5 dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for 72 dollars?

NOTE: Let denote the length of the side of the base and denote the height of the box

Question

A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing 1 dollar/sq ft, and the top is to be constructed of glass costing 5  dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for 72 dollars?

NOTE: Let  denote the length of the side of the base and  denote the height of the box

anonymous · Answer

put $x$ as the length of the square base so the area will be $x^2$ and the cost will be $x^2+5x^2=6x^2$ for the top and bottom.

if you call the height $h$ then the four sides will have area $xh$ for a total of $4xh$ 
therefore you total cost is $6x^2+4xh=72$ 
solve for $h$ and use that in the formula for the volume $$V=x^2h$$

anonymous · Answer

then maximize the volume by finding the derivative, set it equal to zero to get the critical points, and find the max