Your initial design consists of a rectangular prism where the length of the cross area is two times the height. This new design provides additional surface area to accommodate both passengers' arms. After discussing costs with the manufacturing team, you identify the expenses related to the cross area (length × height):
The cost associated to the cross area of the armrest is $0.50/in^2
As the perimeter of the cross area increases, the cost of the armrest is reduced at a rate of $0.67/in.
The base cost to manufacture the armrest is $6.00.

What is the length of the cross area that would minimize the cost of manufacturing?