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Suppose postal requirements are that the maximum of the length plus girth (cross sectional perimeter) of a rectangular package that may be sent is 144 inches. Find the dimensions of such a package with square ends whose volume is to be a maximum.

$$48 \times 24 \times 24$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

03:50

Maximum Volume A rectangul…

04:11

A rectangular package to b…

03:08

03:56

rectangular package to be …

02:59

06:54

03:45

Volume Find the dimensions…

01:16

Postal regulations specify…

So I told that the maximum um what are they? Well it's a metric that the post office uses. They do actually use this um To decide how, you know how big how big of a package you can send. And also like I think it depends on the cost to. So the biggest they say this our value is 144". Now what that is is it's the some of the the height, the width and the length of the package. So assuming it's a rectangular solid. Uh We just some each of these dimensions and they say the maximum of that somehow can be that some can't be any more than 100 and 44 inches. Now in this case we're told egg was be so we have a square cross section here so that we wanna we wanna maximize the volume given this constraint here. And so the volume is you know, A times B times C. Or b squared times see if A equals B. And then our equals two B plus C equals 1 44. So we can solve this for C plug it back into here and we get the volume is to be squared Times The Quantity 1 72 -7. And taking the derivative. This setting people to be one, we get 61 times the quantity 48 -71 and set that equal to zero. So for B one and one very well obviously get is zero. Yeah well that is um you know be a zero then C. Is 144. But we basically have a line and so we have no volume there, so that's the minimum, so we don't want that one. And the other answer is B one is 48 inches, And if we then plug that into here you can see one is also 48", so we actually have a cube. So to uh maximize the volume given this constraint, we wind up with a cube.

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