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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).The United Postal Service requires that for any rectangular package, the sum of its length and girth (cross sectional distance) not exceed 108 inches. Determine the dimensions of the package of largest volume that may be sent satisfying this requirement.

$(36,18,18)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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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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01:56

Use Lagrange multipliers i…

02:59

Maximum Volume A rectangul…

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Find the dimensions of the…

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POSTAL REGULATIONS A recta…

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Postal requirements specif…

and we're told that the post office as a restriction I I call it, see it's not really a cost, it's just a restriction on the size of a package. And what they do is they take the length, width and height, length, width and height of the package and add it all up. And that number needs to be um no bigger than 108. So if we're gonna if we're gonna go with the maximum of that so our constraint is we want that to be 108. You want to maximize the volume such that this is 108. The some of the dimensions are going to be great and so we want to maximize the product of the dimensions which is the volume. So we have g equals v plus λ C -1 away. The compartments with respect to all of the variables with A. B. L. And lambda. So they're gonna make 10, We get four equations and foreign loans. Um And what we find is that what our solution to those four equations and four unknowns is that A B and L. Are all equal to each other? So are ideal package is a cube And has a volume of 36 cubed or 46,600 and 56 cubic inches. Now we can also uh that that I do here I think I saw I solved this for Yeah I eliminated L. Here so this is C. Bar and this is A B. So I solved this for L. No no I solved this for L. And substituted into here. This isn't C. Bar, this is V. Bar. So I saw this for L substituted in there and then plotted it. And again I when I plotted, I can see that I do get a maximum somewhere around in here, which is in the ballpark of 40,000. And you know, 36 looks like A&B are about 36 here. So we can be looking if we would actually maximize V. Bar, we should get the same same result. And that gives us confidence that this is the correct answer because we basically got it in two different ways.

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