Math Physics: Lagrange Multiplier question

In summary, the conversation is about a Lagrange Multiplier problem from the book "Mathematical Methods in the Physical Sciences" by Mary Boas. The problem involves finding the proportions that will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, given the surface area. The volume and surface area equations are provided, along with a picture of the cylinder and cone. The person has started solving the problem but is unsure how to get numerical answers without knowing the surface area. They have solved for the values of r, l, and s in terms of lambda, but are still unsure how to get the proportions. The conversation ends with the person mentioning that they have found a way to
  • #1

Homework Statement


Hello. I've been stuck on a Lagrange Multiplier problem. It's from Mathematical Methods in the Physical Sciences by Mary Boas 3rd edition pg. 222. The question is:

What proportions will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, if the surface area is given?

Then there is a picture of a cylinder with a cone attached to the end. the circular base has radius r, cylinder has length l, and slope of the cone is marked s.

So I've started doing the problem, but something just doesn't seem right. How am I supposed to get an answer if I don't know what the surface area is? I looked in the back of the book and there are numerical answers for r, l, and s. How am I supposed to get actual number answers and not something just in terms of the surface area?

I really want to get this clarified before I go much further because the algebra is absolutely horrendous.

Homework Equations



The volume is V=pi*r^2*l+(1/3)*pi*r^2*(sqrt(s^2-r^2))
and the surface area is SA=pi*r*s+pi*r^2+2*pi*r*l

then to do lagrange multipliers you write F=V+b(SA)
(usually lambda is used instead of b)

The Attempt at a Solution



So then you get use the three partial derivatives and the surface area equation and get a system of 4 equations and solve.

I've solved them out to get values of r, l, and s as functions of lambda (b) and it was a huge mess. Now all that's left is to plug them into the surface area equation and somehow get numerical answers...? I'm really confused.
 
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  • #2
bump. I've solved out the algebra and have r, l, and s in terms of lambda but I don't know how to get the proportions.

also I noticed something...I can get s in terms of r and l in terms of r and s. Dunno if that could help or anything though...
 

1. What is the Lagrange multiplier method in math physics?

The Lagrange multiplier method is a mathematical tool used to find the maximum or minimum value of a function subject to equality constraints. It allows us to incorporate these constraints into the optimization problem and find the optimal solution.

2. When is the Lagrange multiplier method used?

The Lagrange multiplier method is used when we need to optimize a function subject to one or more constraints. It is commonly used in mathematical physics to solve problems involving multiple variables and constraints, such as finding the maximum or minimum value of a function in a given physical system.

3. How does the Lagrange multiplier method work?

The Lagrange multiplier method works by introducing a new variable, called the Lagrange multiplier, to the function being optimized. This new variable is used to incorporate the constraints into the optimization problem, and the optimal solution is found by simultaneously solving the original function and the constraint equations.

4. What are the advantages of using the Lagrange multiplier method?

The Lagrange multiplier method allows us to solve complex optimization problems with multiple variables and constraints. It also provides a systematic approach to incorporating constraints into the optimization problem, making it easier to find the optimal solution.

5. Are there any limitations to the Lagrange multiplier method?

One limitation of the Lagrange multiplier method is that it can only be used for problems with equality constraints. It also requires the constraints to be differentiable, which may not always be the case in real-world applications. In addition, the method can become computationally expensive for problems with a large number of constraints.

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