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

total cost, if the base has side x and the height is h, is $1x^2+5x^2+4\times xh=6x^2+5xh=72$
volume is $V=x^2h$ 
solve for h in the first equation and substitute in the second to get the volume in terms of x alone

anonymous · Answer

typo there, should have been $$1x^2+5x^2+4\times xh=6x^2+4xh=72$$

anonymous · Answer

then 
$$h=\frac{72-6x^2}{4x}$$ and so 
$$V(x)=x^2\times \frac{72-6x^2}{4x}$$ clean this up with some algebra, find the derivative, find the critical points, and that will give a maximimum volume