Lesson 11
External Constraints, Lagrange Multipliers

1) Suppose you've been commissioned to build a square pyramid. ( A square pyramid is a pyramid for which the base is a square, and all triangular faces are congruent isosceles triangles.... like the Great Pyramids.) You are supposed to deliver a pyramid with the largest possible volume, given a fixed surface area (not including the square base). The pyramid can be characterized by its height h and the length of the side of the base, a.

The book says that if we have a functional f of a couple of dependent variables, say y and z, and a constraint equation of the form g = 0, these equations can be used to ensure that the integral of f is an extremum:

For the pyramid problem, try to identify what f, y, z, g, and l are. (We need to know how to apply these equations to real problems!) Explain how you'd use these equations to make the best pyramid for the task.

Note: It might be helpful to know that the volume of a square pyramid is given by V = a²h/3 and that the total surface area of the pyramid (not counting the base) is given by A = 2as where s is (h² + 1/4 a²)^1/2.

2) The family dog needs a new dog yard. Suppose nice fencing costs $10/meter. Your family wants to keep the cost of the new dog yard reasonable and wants the dog to have as much roaming and running area as possible for the cost expended. Because it also has to look good, the dog yard must be 4 sided, with right angles.

Estimate the length l and width w of the dog yard your family can have in order to give the dog as much room as possible. As usual, please explain what you are thinking and doing as you answer this question. If you get stuck, explain what it is that you think you are stuck on.

3) If you have a scenario for which there are 3 dependent variables and 5 constraint equations, what would each of your typical Euler equations consist of?

1. 3 terms that involve partial derivatives of f with respect to a dependent variable and 5 terms that involve 5 (different) l's.
2. 3 terms that involve partial derivatives of f with respect to a dependent variable and 1 term that involves 1 l.
3. 1 term that involves a partial derivative of f with respect to a dependent variable and 5 terms that involve 5 (different) l's.
4. 1 term that involves a partial derivative of f with respect to a dependent variable and 1 term that involves 1 l.

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