The manufacturer wants to triple the volume of the rectangular prism (7*9*4 - length*width*height). Which dimensions should he triple to use the least amount of material? Please explain your reasoning with work provided. Thank You!
~Indian Tech Support
V = l*w*h
A = 2*l*w*h*(1/l + 1/w + 1/h)
Suppose the dimension l is tripled, then the new prism has an amount of material
Al = 6*l*w*h*(1/3l + 1/w + 1/h) = 6*l*w*h*(1/l + 1/w + 1/h - 2/3l) = 3A - 4wh
Similarly for the dimensions w and h, Aw = 3A - 4lh and Ah = 3A - 4lw.
To have the least amount of material for the prism, choose Max(wh, lh, lw) = lw
which implies the dimension h is tripled with the prism having a volume of 7*9*12.