In this assignment we will deal with the solid whose faces are (parts of) the planes given by the equations x = 0, y = 0, z = 0, x + 2y + 2z = 20, 2x + y + 2z = 20, 2x +2y + z = 20, and x + y + z = 12. Another way to look at this solid is as the set of points with coordinates (x, y, z) which satisfy all of the following seven inequalities: x ≥ 0, y ≥ 0, z ≥ 0, x + 2y + 2z ≤ 20, 2x + y + 2z ≤ 20, 2x + 2y + z ≤ 20, and x + y + z ≤ 12.

1. Find the coordinates of all of the vertices of this solid and make as accurate a sketch as you can of it.
2. Find the maximum value of the function f(x, y, z) = 2x − y + z on this solid and determine at which point(s) of the solid this maximum occurs.

Note: In this context the inequalities defining the solid are called linear constraints. Problems involving the optimization of a linear function subject to linear constraints arise often enough to be pretty important in the real world.

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Linear Optimization Questions
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