Solving Optimization Word Problems

You're creating a secret room behind a bookcase for your new mansion. The room is to have a volume of $800\text{ ft}^3$. The width of the base of the room is to be four-fifths the length of the base. Material for the base costs $\$10$ per square foot. Material for the walls costs $\$8$ per square foot. Material for the ceiling costs $\$6$ per square foot. Finally, material for the entrance costs $\$12$ per square foot. Find the minimum cost for constructing this room.

Round your answer to the nearest hundredth.