Answer:
(x,y,z) = (12,12,12) cm
Step-by-step explanation:
The box is assumed to be a closed box.
The surface area of a box of dimension x, y and z is given by
S = 2xy + 2xz + 2yz
We're to minimize this function subject to the constraint that
xyz = 1728
The constraint can be rewritten as
xyz - 1728 = 0
Using Lagrange multiplier, we then write the equation in Lagrange form
Lagrange function = Function - λ(constraint)
where λ = Lagrange factor, which can be a function of x, y and z
L(x,y,z) = 2xy + 2xz + 2yz - λ(xyz - 1728)
We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.
(∂L/∂x) = 2y + 2z - λyz = 0
λ = (2y + 2z)/yz = (2/z) + (2/y)
(∂L/∂y) = 2x + 2z - λxz = 0
λ = (2x + 2z)/xz = (2/z) + (2/x)
(∂L/∂z) = 2x + 2y - λxy = 0
λ = (2x + 2y)/xy = (2/y) + (2/x)
(∂L/∂λ) = xyz - 1728 = 0
We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z
(2/z) + (2/y) = (2/z) + (2/x)
(2/y) = (2/x)
y = x
Also,
(2/z) + (2/x) = (2/y) + (2/x)
(2/z) = (2/y)
z = y
Hence, at the point where the box has minimal area,
x = y = z
Putting these into the constraint equation or the solution of the fourth partial derivative,
xyz - 1728 = 0
x³ = 1728
x = 12 cm
x = y = z = 12 cm.
