$(3)$ We want to construct a rectangular box where the height is three times the width of the base. The material used for the top and bottom of the box costs $15$ dollars per square foot, and the material used for the sides costs $10$ dollars per square foot. If the box must have a volume of $81$ cubic feet, determine the dimensions of the box $($width$,$ length, and height$)$ that will minimize the cost of constructing the box.