Answer:
The dimensions of the box that minimizes the amount of material of construction is
Square base = (5.98 × 5.98) in²
Height of the box = 2.99 in.
Step-by-step explanation:
Let the length, breadth and height of the box be x, z and y respectively.
Volume of the box = xyz = 107 in³
The box has a square base and an open top.
x = z
V = x²y = 107 in³
The task is to minimize the amount of material used in its construction, that is, minimize the surface area of the box.
Surface area of the box (open at the top) = xz + 2xy + 2yz
But x = z
S = x² + 2xy + 2xy = x² + 4xy
We're to minimize this function subject to the constraint that
x²y = 107
The constraint can be rewritten as
x²y - 107 = constraint
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 and y
L(x,y,z) = x² + 4xy - λ(x²y - 107)
We then take the partial derivatives of the Lagrange function with respect to x, y and λ. Because these are turning points, at the turning points each of the partial derivatives is equal to 0.
(∂L/∂x) = 2x + 4y - 2λxy = 0
λ = (2x + 4y)/2xy = (1/y) + (2/x)
(∂L/∂y) = 4x - λx² = 0
λ = (4x)/x² = (4/x)
(∂L/∂λ) = x²y - 107 = 0
We can then equate the values of λ from the first 2 partial derivatives and solve for the values of x and y
(1/y) + (2/x) = (4/x)
(1/y) = (2/x)
x = 2y
Hence, at the point where the box has minimal area,
x = 2y
Putting these into the constraint equation or the solution of the third partial derivative,
x²y - 107 = 0
(2y)²y = 107
4y³ = 107
y³ = (107/4) = 26.75
y = ∛(26.75) = 2.99 in.
x = 2y = 2 × 2.99 = 5.98 in.
Hence, the dimensions of the box that minimizes the amount of material of construction is
Square base = (5.98 × 5.98) in²
Height of the box = 2.99 in.
Hope this Helps!!!
