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How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x+5y+8z=40?

User Shfx
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1 Answer

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Answer:

V = 1600/27 ≈ 59.259... cubic units

Explanation:

You want the volume of the largest cuboid in the first octant with its vertex on the plane x +5y +8z = 40.

Cube

The cuboid having the largest volume for a restriction on area or total linear dimensions will be a cube. The areas of opposite faces are equal, and the lengths of edges in the different dimensions are also equal. A further increase in any dimension will cause others to decrease, reducing the volume.

This describes a problem similar to the given one, but with the cuboid corner on the plane x + y + z = 1. (The total of linear dimensions is 1.) The largest cube will have dimensions x = y = z = 1/3.

Scaled cube

The given plane can be described by the equation ...

x/40 +y/8 +z/5 = 1

This represents a scaled version of the "cube" problem, with different scale factors in the different directions. Effectively, we have ...

  • x' = x/40
  • y' = y/8
  • z' = z/5

and we want to maximize V' = x'y'z'.

Not surprisingly, the maximum volume will be found where ...

x' = y' = z' = 1/3

So, the maximum volume is ...

V' = x'y'z' = (1/3)³ = V(1/40·1/8·1/5) = V/1600 . . . . . . where V=xyz

V = 1600/27 . . . . . . the maximum volume of the cuboid

Lagrange Multipliers

You can arrive at the same conclusion by working the problem using Lagrange multipliers.

Define the Lagrangian as ...

L = xyz + λ(x +5y +8z -40)

We want to set all the partial derivatives to zero:

  • ∂L/∂x = 0 = yz +λ
  • ∂L/∂y = 0 = xz +5λ
  • ∂L/∂z = 0 = xy +8λ
  • ∂L/∂λ = 0 = x +5y +8z -40

Pairs of the first three equations tell you ...

  • x = 5y
  • x = 8z

Then the last equation tells you x + x + x - 40 = 0, or x = 40/3.

The volume is (x)(x/5)(x/8) = 1600/27, as above.

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Additional comment

You will notice that x = 5y = 8z means the total of these terms is equally split between them. This will always be the case for this sort of optimization problem.

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User Job Evers
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