Respuesta :
Answer:
The dimensions (length, width, height) with the smallest possible surface area are
10 ft, 10 ft and 5 ft respectively.
Step-by-step explanation:
The box is an open-at-the-top box.
If the length, width and height of the box are x, z and y respectively,
The surface area of a box of dimension x, y and z open at the top is given by
S(x,y,z) = 2xy + 2yz + xz
We're to minimize this function subject to the constraint that
Volume = 500 ft³
xyz = 500
The constraint can be rewritten as
xyz - 500 = 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 + 2yz + xz - λ(xyz - 500)
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 + z - λyz = 0
λ = (2y + z)/yz = (2/z) + (1/y)
(∂L/∂y) = 2x + 2z - λxz = 0
λ = (2x + 2z)/xz = (2/z) + (2/x)
(∂L/∂z) = x + 2y - λxy = 0
λ = (x + 2y)/xy = (1/y) + (2/x)
(∂L/∂λ) = xyz - 500 = 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) + (1/y) = (2/z) + (2/x)
(1/y) = (2/x)
y = (x/2)
Also,
(2/z) + (2/x) = (1/y) + (2/x)
(2/z) = (1/y)
z = 2y = 2(x/2) = x
Hence, at the point where the box has minimal surface area,
y = (x/2)
z = x
Putting these into the constraint equation or the solution of the fourth partial derivative,
xyz - 500 = 0
(x)(x/2)(x) = 500
x³ = 1000
x = 10 ft
y = (x/2) = 5 ft
z = x = 10 ft.
Hope this Helps!!!
