Optimizing Window Design: Maximizing Area with Fixed Perimeter

In summary, the conversation discusses how to prove that the area of a window with a fixed perimeter is greatest when its breadth equals its greatest height. Two methods are suggested: using a Lagrange multiplier or assuming a constant perimeter and solving for the surface area as a function of one variable. The latter method is found to be successful in this case.
  • #1
Noir
27
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Homework Statement


A window of fixed perimeter is in the shape of a rectangle surmounted by a semi-circle. Prove that its area is greatest when its breadth equals its greatest height.


Homework Equations


SA = lw + (pi*l^2)/4 <--- Thats what I got the surface area to be.
Perimeter = 2w + L(1 + pi / 2)


The Attempt at a Solution


I can solve these problems with numbers, but when it comes to general problems I become unstuck. I tried using the same methord, but it didn't work. Some advice please?

Thanks
 
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  • #2


you need to maximise the surface area with the perimeter constraint and the easiest way would be to use a lagrange multilpier..

otherwise assume a constant value for the perimeter, say p, solve the perimeter eauqtion for l or w, then substitute back into the SA equation and minimise the function of (now) one variable
 
  • #3


Cheers, the second methord worked a treat! I'll look into the lagrange stuff, looks interesting :)
Thanks once again.
 

1. What is the Lagrange multiplier problem?

The Lagrange multiplier problem is a mathematical optimization problem that involves finding the maximum or minimum value of a function subject to a set of constraints. It is named after the mathematician Joseph-Louis Lagrange who first described the method.

2. What is the purpose of using Lagrange multipliers?

Lagrange multipliers allow for the optimization of a function with a set of constraints by converting it into a simpler unconstrained problem. This method is useful in various fields such as economics, physics, and engineering.

3. How does the Lagrange multiplier method work?

The Lagrange multiplier method involves introducing a new variable (the Lagrange multiplier) to the original function and then finding the values of this variable that satisfy the constraints. The optimal solution is then found by solving a system of equations involving the original function and the constraints.

4. What are some common applications of the Lagrange multiplier problem?

The Lagrange multiplier problem has many applications in different fields, such as in economics for maximizing utility subject to a budget constraint, in physics for optimizing energy or force, and in engineering for minimizing costs while meeting certain specifications.

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

One limitation of the Lagrange multiplier method is that it only works for problems with a single objective function and a set of linear constraints. It also relies on the existence of a differentiable objective function, which may not always be the case.

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