Start by finding a function to represent the cost of making the chest...
The length of a cedar chest is twice its width. The cost/dm² of the lid is 4 times the cost/dm² of the rest of the cedar chest. If the volume of the cedar chest is 1440 dm³, find the dimensions so that the cost is a minimum.
Unfortunately, I don't know where to start (didn't see any questions like this at all in my examples or the questions I've solved).
How would I do that? That seems to be my only problem at the moment. Once I can do that, I can pretty much do the rest (differentiate, find where the derivative is 0, then plug back the minimum into the original equations to find the other unknowns). I know that LWH = 1440³ dm, and that L = 2W. Plugging this back into the equation I get 2W²H = 1440 and then I rearrange for but that's about as far as I can get.