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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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

10th Edition
Ron Larson
ISBN: 9781305860919

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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.

Chapter 7.5, Problem 45E, Volume Find the dimensions of a rectangular package of maximum volume that may be sent by a shipping

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=l⋅b⋅h.

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,

∂f∂x=0

Step1: ∂2f∂x2>0 for the above critical point then f attains a minimum.

Step2: ∂2f∂x2<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=96−2y−2z

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

So,

V=xyz=(96−2y−2z)yz=96yz−2y2z−2yz2

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

Vy=96z−4zy−2z2=2z(48−2y−z)

And,

Vz=96y−2y2−4y

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, ...

Single Variable Calculus

Show that a projectile reaches three-quarters of its maximum height in half the time needed to reach its maximu...

Calculus (MindTap Course List)

In Exercises 49-62, find the indicated limit, if it exists. 60. limx4x4x2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Solve the equations in Exercises 126. 14x2=0

Finite Mathematics

Describe the data for a correlational research study. Explain how these data are different from the data obtain...

Statistics for The Behavioral Sciences (MindTap Course List)

Grocery Stores: New ProductsThe Denver Post slated that 80% of all new products introduced in grocery stores fa...

Understanding Basic Statistics

Consider the strength data for beams given in Example 1.2. a. Construct a stem-and-leaf display of the data. Wh...

Probability and Statistics for Engineering and the Sciences

In problems 7-12, evaluate each algebraic expression at the indicated values of the variables. 8. at

Mathematical Applications for the Management, Life, and Social Sciences

For Problems 27-36, simplify each of the algebraic expressions by combining similar terms. 3a22b27a23b2

Intermediate Algebra

Driving Time You need to buy groceries at the supermarket, deposit a check at the credit union, and purchase a ...

Mathematical Excursions (MindTap Course List)

Evaluate the integral. 32. 13e3/xx2dx

Single Variable Calculus: Early Transcendentals

Calculate the total cost, total depreciation, and annual depreciation for the following assets by using the str...

Contemporary Mathematics for Business & Consumers

18. Find subrings and of such that is not a subring of .

Elements Of Modern Algebra

Find the following percentiles for the data listed in Example 4: 55th percentile

Elementary Technical Mathematics

Problems 21 through 24 refer to Figure 20. Figure 20 If DE=28, could ABCFED?

Trigonometry (MindTap Course List)

The foyer planned as an addition to an existing church is designed as a regular octagonal pyramid. Each side of...

Elementary Geometry For College Students, 7e

Evaluating an Improper Integral In Exercises 79-86, determine whether the improper integral diverges or converg...

Calculus of a Single Variable

Evaluating a Definite Integral as a Limit In Exercises 5-10, evaluate the definite integral by the limit defini...

Calculus: Early Transcendental Functions (MindTap Course List)

Consider the inequality 69. Write the statement that results when a 8 is added to each side. b. each side is mu...

Elementary Geometry for College Students

In Exercises 1720, refer to the following experiment: Urn A contains four white balls and six black balls. Urn ...

Finite Mathematics for the Managerial, Life, and Social Sciences

Finding Volume In Exercises 21-26, use a double integral to find the volume of the indicated solid.

Multivariable Calculus

is a(n): ellipsoid elliptic cone elliptic paraboloid elliptic cylinder

Study Guide for Stewart's Multivariable Calculus, 8th

Area: Two circles of radius 3, with centres at (-2,0) and (2,0), intersect as shown in the figure. Find the are...

Calculus

The arc length function for y = x2, 1 x 3 is s(x) = a) 1x1+4x2dx b) 1x1+4t2dt c) 1x1+t4dt d) 1x1+4t4dt

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric f...

Calculus (MindTap Course List)

Express each volume as indicated. Round each answer to the same number of significant digits as in the original...

Mathematics For Machine Technology

Suppose that an individual is randomly selected from the population of all adult males living in the United Sta...

Introduction To Statistics And Data Analysis

Data for two variables, x and y, follow. xi 1 2 3 4 5 yi 3 7 5 11 14 a. Develop the estimated regression equati...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Discovery and Writing Summarize the general strategy used to solve application problems in this section.

College Algebra (MindTap Course List)

Explain the basic distinction between probability sampling methods and nonprobability sampling methods and reco...

Research Methods for the Behavioral Sciences (MindTap Course List)

Describe the difference between inductive and deductive reasoning and give an example of each.

Research Methods for the Behavioral Sciences (MindTap Course List)

The age composition of the population of the United States in the year 2012 is given in Figure 4.67. aFind the ...

Mathematics: A Practical Odyssey

The weekly demand for a product is believed to be normally distributed. Use a goodness of fit test and the foll...

Statistics for Business & Economics, Revised (MindTap Course List)

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