A container manufacturer plans to make rectangular boxes whose bottom and top measure $4$x by $3$x$.$ The container must contain $48$in$^3.$ The top and the bottom will cost $$1\mathrm{.}80$ per square inch, while the four sides will cost $$3\mathrm{.}70$ per square inch.

What should the height of the container be so as to minimize cost? Round your answer to the nearest hundredth.