I need some help with a problem involving the optimization of area.
This is the prompt to the problem:
"A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides will be made of material costing $1 per square foot, and the top is to be constructed of glass costing $5 per square foot. What are the dimensions of the box of greatest volume than can be constructed for $72?"
I have worked several optimization problems both involving cost and not but in this problem I am given the cost and not the area. I understand the process of optimization: Construct an equation that models the problem (hopefully with a sketch), use the given value in its respective equation and solve for one of the variables, plug this quantity into the main equation and solve for the other variable.
I guess I should start with checking if my equation is correct.
Surface Area = sides + top + bottom
SA = 4xy + x^2 + x^2 Volume = x^2*y
Cost = 4xy(1) + x^2(5) + x^2(1) Given value : $72 (max cost)
Cost = 4xy + 5x^2 + x^2
With problems like this I usually just take the derivative and doing that gave me the wrong dimensions. So going back I think that I need to set the cost equation equal to 72. However that would then incorporate implicit differentiation into an optimization problem and I have never done that before, just implicit by itself. Am I headed in the right direction and if so then what?