Find the maximum volume of a box inscribed in the tetrahedron bounded by the coordinate planes and the plane x + 1 3 y + 1 8 z = 1.

Considering the spaces in the coefficients, I'm guessing the plane has equation

Let be the vertex of the box that lies in this plane. The box has volume

and we want to maximize this subject to the plane equation. Using the method of Lagrange multipliers, the Lagrangian is

with partial derivatives (set equal to 0)

We have

(we ignore and because those values correspond to a degenerate box with zero volume)

So we get a maximum volume of .

step-by-step explanation:

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