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 ...
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.
__
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.
<95141404393>