Suppose you have a rectangular box in the first {\bf octant} (a really cool word! look it up if you don't know it!), such that it has a corner on the origin, sides lying along the positive $x$, $y$, and $z$ axes (or parallel to them), and another corner, opposite the corner at the origin, lying on the plane: $$2x + 4y + 5z = 20$$ What are the dimensions of the rectangle with the greatest volume?
I know this is a Lagrange Multiplier problem, as we are given a clear restraint and can create a function for the volume. I am just confused as to how this would be set up in terms of the restraint function.
The method of Lagrange multipliers, is $\nabla f(x,y,z) = \lambda \nabla g(x,y,z)$, where $f$ is your volume and $g$ is the constraint to the plane. The gradient of the constraint is: $$<2,4,5>$$ and the gradient of the volume $f(x,y,z) = xyz$ is: $$<yz,xz,xy>$$ So the system of equations that you need to solve is: $$yz = 2\lambda$$ $$xz = 4\lambda$$ $$xy = 5\lambda$$ $$2x+4y+5z=20$$ First assume that none are zero, as if one of them is zero, then the volume is zero, obviously not optimal. Substitute the first two equations into the third and fourth to get: $$8\lambda^2/z^2 = 5\lambda$$ $$16\lambda/z + 5z = 20$$ And so: $$\lambda= 5z^2/8$$ Sub this into the fourth equation to get the final answer of: $$x = 10/3, y = 5/3, z = 4/3$$ And a volume of $200/27$