Let x =lenght, y = width, and z =height
The volume of the box is equal to V = xyz
Subject to the surface area
S = 2xy + 2xz + 2yz = 64
= 2(xy + xz + yz)
= 2[xy + x(64/xy) + y(64/xy)]
S(x,y)= 2(xy + 64/y + 64/x)
Then
Mx(x, y) = y = 64/x^2
My(x, y) = x = 64/y^2
y^2 = 64/x
(64/x^2)^2 = 64
4096/x^4 = 64/x
x^3 = 4096/64
x^3 = 64
x = 4
y = 64/x^2
y = 4
z= 64/yx
z= 64/16
z = 4
Therefor the dimensions are cube 4.
