Question

Use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane x + 3y + 2z = 9?

Answer 1

To determine the volume of the largest rectangular box within the first octant and bound by the plane x + 3y + 2z = 9, we apply the method of Lagrange multipliers to maximize the volume function V = xyz subject to the given constraint, and solve for x, y, z values.

Step-by-step explanation:

To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane x + 3y + 2z = 9, we use Lagrange multipliers. First, we define the volume V of the rectangular box with sides x, y, z as V = xyz. We want to maximize V subject to the constraint g(x, y, z) = x + 3y + 2z - 9 = 0.

We set up the system of equations given by the method of Lagrange multipliers:

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• λ * ∇g(x, y, z) = ∇V
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• x + 3y + 2z = 9

Using partial derivatives, we get:

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• λ = yz
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• 3λ = xz
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• 2λ = xy

By solving this system, we find the relationship between x, y, and z. Finally, we substitute these back into the volume equation to find the maximum volume.

To illustrate the maximum volume V, consider the system:

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• y/z = 1/3
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• x/z = 1/2
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• x/y = 2/3

After determining the values of x, y, and z that satisfy these ratios and the plane equation, we calculate V = xyz for the largest rectangular box that fits the given condition.