Probem12+Elise,+Mike+J.

very good! (ms sweeney) 12. V=lw^2=32,000 cm^3 l=32,000/w (and I think you mean l = 32,000/w²

(just point out that THIS is your objective function, which you are trying to Mimimize) Minimize: A= 4lw+w^2 A=4(32000/w) + w^2, w>0 A=128000/w + w^2

A'=-128000/w^2 + 2w=0 2w=128000/w^2 2w^3=128000 w^3=64000 w=40

40x40=1600 32000/1600=20

Therefore, to minimize the amount of material neede, the dimensions of the box should be 40cm x 40cm x 20cm 

The domain for this function would be (0,infinity). When we take the limit of the objective function, which we are trying to minimize, as the height approaches infinity and the base sides approach 0, the total area approaches infinity. And as the height approaches zero and the base sides approach infinity, the total surface area of the box once again approaches infinity. This means that the behavior at the critical value must be the absolute minimum for the objective function.