A company is designing a new product and needs to transport it in a rectangular box. The box will be shipped by a courier that charges based on the box's dimensions. The surface area of the box is 384 square inches, and the sum of the lengths of its 12 edges is 112 inches. If the courier charges $0.05 per cubic inch for the volume of the box, how much will it cost to ship the box? Let's think step by step and output the final answer within \boxed{}.
user