An open$-$top rectangular box is being constructed to hold a volume of $150i{n}^3.$ The base of the box is made from a material costing $6$ cents$/$in ${?}^2.$ The front of the box must be decorated, and will cost $11$ cents$/$in ${?}^2.$ The remainder of the sides will cost $3$ cents$/$in ${?}^2.$

Find the dimensions that will minimize the cost of constructing this box.

Front width: $$ in$.$

Depth: $$ in$.$

Height: in$.$