Question 1035166
The volume of the box must be 12 m3. 
The cost per square meter of material for the base is $0.50, for the sides $0.20, and for the lid $0.10.
 If the total cost of materials is a minimum, then the dimensions (in meters) of the box are
:
let x = the side of the square base
then
x^2 = the area of the base and the lid
and the volume is to be 12 cu/m. therefore:
12/x^2 = height of the box 
:
Cost = base area + side areas + lid area
C = .5x^2 + .2(4*x*{{{12/x^2}}}) + .1x^2
cancel x
C = .6x^2 + .2(4*{{{12/x}}})
C = .6x^2 + {{{9.6/x}}}
Graphically, we can see minimum cost occurs when x = 2
{{{ graph( 300, 200, -3, 5, -5, 20, .6x^2+(9.6/x)) }}}
The dimensions
L=2; 
W=2
H = 12/2^2 = 3 m