A box of constant volume c is to be twice long as it is wide . the material on the top and four sides cost three times as much per square metre as that in the bottom. what are most economic dimensions ?

Answer 1

w * 2w * h = c
So, h = c/(2w^2)

So if the bottom costs $1/m^2, the total cost is

C(w,h) = w*2w + 3*w*2w + 3*2(wh + 2wh)
= 7w^2 + 18wh
C(w) = 7w^2 + 18w*c/(2w^2)
= 7w^2 + 9c/w

dC/dw = 14w - 9c/w^2
dC/dw=0 when (14w^3-9c) = 0
That is, when w = ∛(9c/14)
and so on to the length and height.