A rectangular building is to have a volume of $8000ft^3$. Annual heating and cooling costs will amount to $\$2/ft^2$ for its top, front, and back, and $\$4/ft^2$ for the two end walls. What dimensions of the building would minimize these annual costs?
I have attached my work so far. However, I end up with a system that I don't know how to solve. Is there any easy way to solve this that I'm missing?
Solve your first three equations for $\lambda$ and make two equations out of those three equations by eliminating $\lambda$.
Then you have three equations and three unknown which you can solve for the dimensions.