The cabinet that will enclose the Acrosonic model D loudspeaker system will be rectangular and will have an internal volume of 1.8 ft3. For aesthetic reasons, it has been decided that the height of the cabinet is to be 1.4 times its width. If the top, bottom, and sides of the cabinet are constructed of veneer costing 35¢ per square foot and the front (ignore the cutouts in the baffle) and rear are constructed of particle board costing 19¢ per square foot, what are the dimensions of the enclosure that can be constructed at a minimum cost?

This is a minimization problem. Is this a minimization of area or volume? What I have so far.

V=LWH. 1.8 = x^2(1.4x) my constraint? I’m assuming the bottom is a square so LW is both x and H is 1.4x.

Cost = 2bottoms * .35 + 2side *.35 + 2side *.19 = .35(2x^2) + .35((2(x*1.4x)) +.19((2(x*1.4x))

This problem is a little different than my previous minimization maximization problems. I can’t seem to combine them to find a derivative. Please help thank you!