Question

A box has a bottom with one edge 8 times as long as the other. if the box has no top and the volume is fixed at v, what dimensions minimize the surface area?

Answer 1

 height: h=V/w/(9w)= V/(9w^2) 
surface area: w*9w+ h*(20w)= 9w^2 + 20wV/(9w^2) = 9w^2 + 20/9 V/w 
minimum at: 
w = cube root of (10/81 V) 
dimensions: 
w, 9w, V/(9w^2)