A box has a bottom with one edge 7 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? Please help me find the COMPLETE dimenssion of the box.

Answer 1

length L
width w
depth d
vol = V = Lwd
area = A = 2wd+2Ld+Lw
but L = 7w
so V = 7w^2d
and A = 2wd+14wd+7w^2
V = 7w^2d = constant
d =V/7w^2
and A = 16w(V/7w^2) + 7w^2
A = (16/7)V/w + 7w^2
dA/dw = 0 at max = (16V/7) (-1/w^2)+14w
(16V/7) = 14 w^3
w = [(8/49)V]^(1/3)
w = .547 V^(1/3)
go back and get L and d in terms of V^(1/3)
CHECK MY ARITHMETIC!