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

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Chapter 7.5, Problem 47E
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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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Cost A manufacturer makes an open-top wooden crate having a volume of 18 cubic feet. Material costs are $0.20 per square foot for the base and $0.15 per square foot for the sides. Find the dimensions that minimize the cost of each crate. What is the minimum cost?

Chapter 7.5, Problem 47E, Cost A manufacturer makes an open-top wooden crate having a volume of 18 cubic feet. Material costs

To determine

To calculate: The dimensions that minimize the cost of each crate and the minimum cost. If The volume of an open-top wooden crate is 18 cubic feet. the cost of material is $0.20 per square foot for the base and $0.15 per square foot for the side.

Explanation

Given Information:

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.

Since, the volume of an open-top wooden crate is 18 cubic feet. so,

xyz=18

It can be written as z=18xy.

The cost of each crate is

C=0.2xy+0.15(2)xz+0.15(2)yz=0.2xy+0.3xz+0.3yz=0.2xy+0.3x(18xy)+0.3y(18xy)=0.2xy+5.4y+5.4x

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

Cx=0.2y−5

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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
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Scores on a quiz were normally distributed and had a mean of 10 and standard deviation of 3. For each of the fo...

Essentials Of Statistics

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or gi...

Single Variable Calculus

The German mathematician Karl Weierstrass 18151897 noticed that the substitution t=tan(x/2) will convert any ra...

Calculus (MindTap Course List)

23/2

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

Evaluate the expression sin Exercises 116. 22212424

Finite Mathematics

A sample of n=16 scores has a mean of M=56and a standard deviation of s=12 . a. Explain what is measured by the...

Statistics for The Behavioral Sciences (MindTap Course List)

For Problems 9-19. please provide the following information. (a) What is the level of significance? State the n...

Understanding Basic Statistics

In Problems 7-34, perform the indicated operations and simplify. 19.

Mathematical Applications for the Management, Life, and Social Sciences

For Problems 3-11, simplify each numerical expression. 12(37)25(217)

Intermediate Algebra

Use the probability demonstrator, in the left margin, to work Excursion. Explain why the greatest number of bal...

Mathematical Excursions (MindTap Course List)

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribu...

Probability and Statistics for Engineering and the Sciences

Trace or copy the graph of the given function .f. (Assume that the axes have equal scales.) Then use the method...

Single Variable Calculus: Early Transcendentals

Write each number in decimal form: 57.7103

Elementary Technical Mathematics

Review Exercises State whether the statements in Review Exercises 1 to 12 are always true A, sometimes true S, ...

Elementary Geometry For College Students, 7e

Let and be integers with and with . Use this result to find the unique quotient and remainder as described by ...

Elements Of Modern Algebra

Solve the following word problems for the unknown. Round decimals to hundredths and percents to the nearest ten...

Contemporary Mathematics for Business & Consumers

Use your graphing calculator to convert the complex number to trigonometric form in the following problems. Pro...

Trigonometry (MindTap Course List)

Using Rolle's Theorem In Exercises 11-24, determine whether Rolle's Theorem can be applied to f on the closed i...

Calculus of a Single Variable

Verifying an Integration Rule the rule by differentiating. Let a 0. dua2u2=arcsinua+C

Calculus: Early Transcendental Functions (MindTap Course List)

Note: Exercises preceded by an asterisk are of a more challenging nature. Given:OwithOECDCD=OCFind:mCF

Elementary Geometry for College Students

In Exercises 2326, find an equation of the line that passes through the point and has the indicated slope m. (3...

Finite Mathematics for the Managerial, Life, and Social Sciences

Given the Taylor Series , a Taylor series for ex/2 is:

Study Guide for Stewart's Multivariable Calculus, 8th

Evaluating a Definite Integral In Exercises 59-66, evaluate the definite integral. 0sin5xdx

Calculus (MindTap Course List)

Sometimes, Always, or Never: If for all n and both {an} and {cn} converge, then{bn} converges.

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

Evaluating a function In Exercises 11 and 12 evaluate the vector-valued function at each given value of t. r(t)...

Multivariable Calculus

Finding an Indefinite Integral In Exercises 5-28, find the indefinite integral. 954xdx

Calculus

Two new assembly methods are tested and the variances in assembly times are reported. Use = .10 and test for e...

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

Fill in the blanks. Natural number greater than 1 that are not prime are called _________ numbers.

College Algebra (MindTap Course List)

Winter visitors are extremely important to the economy of Southwest Florida. Hotel occupancy is an often-report...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Describe why you might be cautious about using the Internet to find answers to medical questions.

Research Methods for the Behavioral Sciences (MindTap Course List)

Cohens kappa was developed to deal with a particular problem with measuring inter-rater reliability. Identify t...

Research Methods for the Behavioral Sciences (MindTap Course List)

In each of the following exercises, the value on the left must be multiplied by one of the following numbers: 0...

Mathematics For Machine Technology

Suppose that we define the following events: C = event that a randomly selected driver is observed to be using ...

Introduction To Statistics And Data Analysis

This is an exercise in buying a car. It involves choosing a home and selecting the homes financing. Write a pap...

Mathematics: A Practical Odyssey

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