A box with an open top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the box has the least possible surface area, find its dimensions.

Question

A box with an open top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the box has the least possible surface area, find its dimensions.

Height =  (include units)

Length of base =  (include units)

Answer 1

Since no top, and square box, so l=w,

S = l² + 4lh

Minimize S subject to V = lwh = l²h = 4

Then, h = 4/l²

Minimize S = l² + 4lh = l² + 4l*4/l² = l² + 16/l

dS/dl = 2l - 16/l²

dS/dl = 0 at the critical point, so 2l - 16/l² = 0 or 2l = 16/l² or 2l³ = 16 or l = 2

d²S/dl² = 2 + 32/l³ > 0 for all l > 0, so this is the minimum.

l = 2 m

h = 4/l² = 4/2² = 1 m

Answer 2

Lwh=4cu/m

4^(1/3)=1.586......=1586......cm^3.