13. Find the minimum volume of a box that must be constructed using 24 square meters of materials for the six sides. Let (x, y, z) = xyz be the volume equation. Find the constraint equation and use the Method of Lagrange Multipliers.
Here, we need to find the minimum volume of a box that must be constructed using 24 squre meters of materials for the six sides.
The six sides are written to be 2xy+2yz+2zx and as per the above condition, 2xy+2yz+2zx = 24 i.e.; xy+yz+zx = 12. ----(1)
Let g(x, y, z) = xy+yz+zx.
We need to mnimize the function, V(x, y, z) = xyz.
Here, we will be using the method of Lagrange Multipliers.
So, by the Method of Lagrange Multipliers,
We need to solve,
So,
------(*)
Here, first let us consider,
Hence, there are three possibilities, λ = 0 or y = 0 or x = z.
But if λ = 0, then inserting it to (*) leads to yz = zx = xy = 0, thus 2 × 0 + 2 × 0 + 2 × 0 − 24 = 0, which leads to −24 = 0, 1 which is impossible. Thus, λ ≠ 0. Also, y = 0 cannot happen since the width of the box must be positive. Therefore, we must have x = z.
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