# Thread: Maximization using Lagrange multipliers, word problem

1. ## Maximization using Lagrange multipliers, word problem

A manufacturer makes two models of an item, standard and deluxe. It costs \$40 to manufacture the standard model and \$60 for the deluxe. A market research firm estimates that if the standard model is priced at $x$ dollars and the deluxe at $y$ dollars, then the manufacturer will sell $500(y−x)$ of the standard items and $45000+500(x−2y)$ of the deluxe each year. How should the items be priced to maximize profit?

Seems like they want to solve this using Lagrange multipliers (is it even possible?)

$P(x,y)=500(y-x)(x-40)+(45000+500(x-2y))(y-60)$

What could be the constraint?

2. ## Re: Maximization using Lagrange multipliers, word problem

You can solve this without using Lagrange multipliers. Just set the Gradient of the profit function to 0 and solve for x and y. Then just check to make sure you found a maximum profit not a minimum one.

3. ## Re: Maximization using Lagrange multipliers, word problem

But how would I do that with Lagrange multipliers? It seems like it's easy to solve without them, but it was in the chapter about Lagrange multipliers. So I don't know...

4. ## Re: Maximization using Lagrange multipliers, word problem

You could choose the constraints that you must sell a non-negative number of items. So $x$ and $y$ must satisfy that each of those market estimate numbers are non-negative.