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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 3

Extrema

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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rectangular package to be …

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A rectangular package to b…

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The sum of the length and …

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Maximum Volume A rectangul…

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Postal Regulations The U.S…

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For this problem, we are told that the United Postal Service requires that for any rectangular package, the sum of its length and its girth, the prosection. All distance must not exceed 108 in were then asked to determine the dimensions of the package of largest volume that may be sent satisfying this requirement. So the length here with the way that I've assigned the variables we have, the length is going to be said and then the dearth is going to be uh two Y plus two X, Y plus two X. We know that that must be equal to 108. Um Then we want to maximize the volume which is just going to be X, Y. Z. So we have F equals X, Y. Z. And then we have our constraint. So the way that we will be able to find this is using the method of Lebron's multipliers. Specifically, we'll say that we have the left hand side of that constraint is going to be G. We want delta F. Two equal lambda time, or the gradient of F. To be equal to lambda times the gradient of G. And also we need to satisfy the constraint. So from the gradients we'll have that with respect to X. You need to have that. Why is that? That's why Zed must equal lambda times uh would be land at times too. So it must equal to lambda with respect to why we'll have that X said must equal uh to land again and with respect to Z will have Xy must equal lambda. And then in addition we have the constraint equation, Z plus two, Y plus two, X equals 108. So we have a system of four equations and four unknowns here. The solution, which maximizes the volume will be λ equals 324 X equals 18, y equals 18, and z equals 36.

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