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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).An athletic field is a rectangular region with a semicircle on each end. If the perimeter of the field is to be 250 yards, determine its dimensions if the field if its rectangular portion is to have maximum area.

Length $=125 / 2$ yd. radius $=\frac{125}{2 \pi} \mathrm{yd}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Use the method of Lagrange…

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And this problem we have an athletic field and it's a rectangle with semicircular ends like this. And we want the perimeter to be 150 yards. So maybe we have fence that we need to put around here are some kind of barrier that we need to put around here and we have 150 yards of it. But we want to make it so that we maximize the area of this rectangle here. Now the perimeter we have a perimeter of a whole circle that has a diameter of W. So that's winds up being pie W. Then we have L. Plus another Elsa to L. And that needs to be 2 50. The area of this rectangle is at all times W. So our augmented function with the lagrange multiplier is a the area plus lambda times P -250. All right. So We have three variables l. W. And lambda. Taking with respect to W. We get L. Plus lambda pi equals zero expected L. We get W Plus two lambda equals zero. Expect the land that we get our constraint equation back. P course to 50. Now we have three linear equations and three unknowns. We can find we saw them and we get W. Goes 1 25 over pie L equals 1 25/2. And those are in the yards. So we have L. Is a um Uh huh. I use the capital L. Here instead of a lower case. L should fix that. Um You know this? It's a little wider than it is. You know, it's about 2/3 another racial aspect ratio of this. But we can then figure out, plug that back into here and figure out what is our area here. And that comes out to be about 2000, about 25,000 square yards of area of our field and our rectangle here. And we don't care, for whatever reason, we don't care about the area here.

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