   Chapter 7.5, Problem 45E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Volume Find the dimensions of a rectangular package of maximum volume that may be sent by a shipping company, assuming that the sum of the length and the girth (perimeter of a cross section) cannot exceed 96 inches. To determine

To calculate: The dimensions of a rectangular package such that the volume is maximum and whose sum of the length and the girth cannot exceed 96 inches.

Explanation

Given Information:

A rectangular package has a sum of length and girth that does not exceed 96 inches and the provided figure is,

Formula used:

Consider a package (rectangular) of the following dimensions,

Then the volume is V=lbh.

Also, if there is a function f in a variable x, it’s critical point can be found by taking a derivative and setting it equal to zero,

fx=0

Step1: 2fx2>0 for the above critical point then f attains a minimum.

Step2: 2fx2<0 for the above critical point then f attains a maximum.

Calculation:

Let the length of the rectangular package be x, width be y and height be z.

A rectangular package has a sum of length and girth that does not exceed 96 inches. So,

Length+girth=96x+2y+2z=96x=962y2z

Now, the volume of the package is V=xyz.

So,

V=xyz=(962y2z)yz=96yz2y2z2yz2

Now find partial derivative of V with respect to y and z,

Vy=96z4zy2z2=2z(482yz)

And,

Vz=96y2y24y

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