An open top box with a square base is being constructed with two different types of material. The material for the lateral faces costs $4$ dollars per square inch. The material for the base costs $8$ dollars per square inch. The goal is to minimize the cost while obtaining a certain volume $V.$ Let $x$ denote the side length of the base and $y$ denote the height of the box. Write the objective function for this optimization problem in terms of $x$ and $y.$