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

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Chapter 7, Problem 79RE
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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 has an order for 1000 wooden benches that can be produced at two locations. Let x1 and x2 be the numbers of wooden benches produced at the two locations. The cost function is modeled by

C = 0.25 x 1 2 + 10 x 1 + 0.15 x 2 2 + 12 x 2 .

Use Lagrange multipliers to find the number of wooden benches that should be produced at each location to minimize the cost.

To determine

To calculate: The number of wooden benches that should be produced at each location to minimize the cost by application of Lagrange multipliers when an order for 1000 wooden benches has to be produced at two different locations where x1 and x2 is assumed to be the number of wooden benches produced at two different locations and the cost function is given by,

C=0.25x12+10x1+0.15x22+12x2

Explanation

Given Information:

An order for 1000 wooden benches has to be produced at two different locations where x1 and x2 is assumed to be the number of wooden benches produced at two different locations and the cost function is given by the function,

C=0.25x12+10x1+0.15x22+12x2

Formula used:

To find the maximum or minimum values of function f follow the steps as,

Step 1: Solve the provided system of equations as,

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Step 2: Next calculate the function f at each of the solution point obtained from first step where the greatest value obtained gives the maximum of function f subject to constraint g(x,y)=0 and the least value gives the minimum of function f subject to constraint g(x,y)=0.

Calculation:

Here the function is C=0.25x12+10x1+0.15x22+12x2 and the subject to constraints is,

x1+x2=1000

Then, let f(x1,x2)=0.25x12+10x1+0.15x22+12x2 and g(x1,x2)=x1+x2−1000.

Then define a new function F as,

F(x1,x2,λ)=f(x1,x2)−λg(x1,x2)=0.25x12+10x1+0.15x22+12x2−λ(x1+x2−1000)

Now solve the equation by finding the partial derivatives of F with respect to x1,x2 and λ as,

Fx1(x1,x2,λ)=0.50x1+10−λFx2(x1,x2,λ)=0.30x2+12−λFλ(x1,x2,λ)=−(x1+x2−1000)

Then set each of the obtained partial derivatives equal to 0 as,

0

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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 P-14ESect-7.2 P-15ESect-7.2 P-16ESect-7.2 P-17ESect-7.2 P-18ESect-7.2 P-19ESect-7.2 P-20ESect-7.2 P-21ESect-7.2 P-22ESect-7.2 P-23ESect-7.2 P-24ESect-7.2 P-25ESect-7.2 P-26ESect-7.2 P-27ESect-7.2 P-28ESect-7.2 P-35ESect-7.2 P-36ESect-7.2 P-37ESect-7.2 P-38ESect-7.2 P-39ESect-7.2 P-40ESect-7.2 P-41ESect-7.2 P-42ESect-7.2 P-43ESect-7.2 P-44ESect-7.2 P-45ESect-7.2 P-46ESect-7.2 P-47ESect-7.2 P-48ESect-7.2 P-49ESect-7.2 P-50ESect-7.2 P-51ESect-7.3 P-1CPSect-7.3 P-2CPSect-7.3 P-3CPSect-7.3 P-4CPSect-7.3 P-5CPSect-7.3 P-1SWUSect-7.3 P-2SWUSect-7.3 P-3SWUSect-7.3 P-4SWUSect-7.3 P-5SWUSect-7.3 P-6SWUSect-7.3 P-7SWUSect-7.3 P-8SWUSect-7.3 P-9SWUSect-7.3 P-10SWUSect-7.3 P-1ESect-7.3 P-2ESect-7.3 P-3ESect-7.3 P-4ESect-7.3 P-5ESect-7.3 P-6ESect-7.3 P-7ESect-7.3 P-8ESect-7.3 P-9ESect-7.3 P-10ESect-7.3 P-11ESect-7.3 P-12ESect-7.3 P-13ESect-7.3 P-14ESect-7.3 P-15ESect-7.3 P-16ESect-7.3 P-17ESect-7.3 P-18ESect-7.3 P-19ESect-7.3 P-20ESect-7.3 P-21ESect-7.3 P-22ESect-7.3 P-23ESect-7.3 P-24ESect-7.3 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P-20QYSect-7.5 P-21QYSect-7.5 P-22QYSect-7.5 P-23QYSect-7.5 P-24QYSect-7.5 P-25QYSect-7.5 P-26QYSect-7.6 P-1CPSect-7.6 P-2CPSect-7.6 P-3CPSect-7.6 P-4CPSect-7.6 P-1SWUSect-7.6 P-2SWUSect-7.6 P-3SWUSect-7.6 P-4SWUSect-7.6 P-5SWUSect-7.6 P-6SWUSect-7.6 P-7SWUSect-7.6 P-8SWUSect-7.6 P-9SWUSect-7.6 P-10SWUSect-7.6 P-1ESect-7.6 P-2ESect-7.6 P-3ESect-7.6 P-4ESect-7.6 P-5ESect-7.6 P-6ESect-7.6 P-7ESect-7.6 P-8ESect-7.6 P-9ESect-7.6 P-10ESect-7.6 P-11ESect-7.6 P-12ESect-7.6 P-13ESect-7.6 P-14ESect-7.6 P-15ESect-7.6 P-16ESect-7.6 P-17ESect-7.6 P-18ESect-7.6 P-19ESect-7.6 P-20ESect-7.6 P-21ESect-7.6 P-22ESect-7.6 P-23ESect-7.6 P-24ESect-7.6 P-25ESect-7.6 P-26ESect-7.6 P-27ESect-7.6 P-28ESect-7.6 P-29ESect-7.6 P-30ESect-7.6 P-31ESect-7.6 P-32ESect-7.6 P-33ESect-7.6 P-34ESect-7.6 P-35ESect-7.6 P-36ESect-7.6 P-37ESect-7.6 P-38ESect-7.6 P-39ESect-7.6 P-40ESect-7.6 P-41ESect-7.6 P-42ESect-7.6 P-43ESect-7.6 P-44ESect-7.6 P-45ESect-7.7 P-1CPSect-7.7 P-2CPSect-7.7 P-3CPSect-7.7 P-1SWUSect-7.7 P-2SWUSect-7.7 P-3SWUSect-7.7 P-4SWUSect-7.7 P-5SWUSect-7.7 P-6SWUSect-7.7 P-7SWUSect-7.7 P-8SWUSect-7.7 P-1ESect-7.7 P-2ESect-7.7 P-3ESect-7.7 P-4ESect-7.7 P-5ESect-7.7 P-6ESect-7.7 P-7ESect-7.7 P-8ESect-7.7 P-9ESect-7.7 P-10ESect-7.7 P-11ESect-7.7 P-12ESect-7.7 P-13ESect-7.7 P-14ESect-7.7 P-15ESect-7.7 P-16ESect-7.7 P-17ESect-7.7 P-18ESect-7.7 P-19ESect-7.7 P-20ESect-7.7 P-21ESect-7.7 P-22ESect-7.7 P-23ESect-7.7 P-24ESect-7.7 P-25ESect-7.7 P-26ESect-7.7 P-27ESect-7.7 P-28ESect-7.7 P-29ESect-7.7 P-30ESect-7.8 P-1CPSect-7.8 P-2CPSect-7.8 P-3CPSect-7.8 P-4CPSect-7.8 P-5CPSect-7.8 P-6CPSect-7.8 P-1SWUSect-7.8 P-2SWUSect-7.8 P-3SWUSect-7.8 P-4SWUSect-7.8 P-5SWUSect-7.8 P-6SWUSect-7.8 P-7SWUSect-7.8 P-8SWUSect-7.8 P-9SWUSect-7.8 P-10SWUSect-7.8 P-11SWUSect-7.8 P-12SWUSect-7.8 P-13SWUSect-7.8 P-14SWUSect-7.8 P-15SWUSect-7.8 P-16SWUSect-7.8 P-1ESect-7.8 P-2ESect-7.8 P-3ESect-7.8 P-4ESect-7.8 P-5ESect-7.8 P-6ESect-7.8 P-7ESect-7.8 P-8ESect-7.8 P-9ESect-7.8 P-10ESect-7.8 P-11ESect-7.8 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Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Finite Mathematics and Applied Calculus (MindTap Course List)

Average Cost A clothing manufacturer finds that the cost of producing x shirts is 500 + 6x + 0.01x2 dollars. (a...

Precalculus: Mathematics for Calculus (Standalone Book)

Transformations What are the three basic types of function transformations?

Calculus: Early Transcendental Functions

By any method, determine all possible real solutions of each equation in Exercises 1330. Check your answers by ...

Applied Calculus

The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the len...

Single Variable Calculus: Early Transcendentals, Volume I

Determine whether the series is convergent or divergent. 19. n=1n3n4+4

Multivariable Calculus

Find y by implicit differentiation. 38. x3 y3 = 7

Calculus: Early Transcendentals

In a sample of n 6 scores, five of the scores are each above the mean by one point. Where is the sixth score l...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

A researcher is studying the effects of the college experience on attitudes, values, and behaviors and is compa...

Essentials Of Statistics

38 Evaluate the iterated integral. 00101z2zsinxdydzdx

Calculus (MindTap Course List)

Given that the point P(2, 4) lies on the line 2x + ky = 8, find k.

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

Expand Your knowledge: Split Stem The Boston Marathon is the oldest and best-known U.S. marathon. It covers a r...

Understanding Basic Statistics

A researcher conducts a study comparing two different treatments with a sample of n = 16 participants in each t...

Statistics for The Behavioral Sciences (MindTap Course List)

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard funct...

Single Variable Calculus

In Problems 51 and 52, rationalize the numerator of each fraction and simplify. 52.

Mathematical Applications for the Management, Life, and Social Sciences

In Exercises 87 to 90, determine whether the given sets are equal. A=2n1nNB={ (n(n+12)nN }

Mathematical Excursions (MindTap Course List)

For problems 15-26, simplify each numerical expression. Be sure to take advantage of the properties whenever th...

Intermediate Algebra

Convert the expressions in Exercises 6584 to power form. 3x453x+43xx

Finite Mathematics

Specimens of three different types of rope wire were selected, and the fatigue limit (MPa) was determined for e...

Probability and Statistics for Engineering and the Sciences

Evaluate the integral using integration by parts with the indicated choices of u and dv. 2. xlndx;u=lnx,dv=xdx

Single Variable Calculus: Early Transcendentals

7. In addition and subtraction of fractions, the most efficient common denominator is the ___________ common de...

Contemporary Mathematics for Business & Consumers

In RSTV, mS=57. Which diagonal (VSorRT) would have greater length?

Elementary Geometry For College Students, 7e

Subtract: (+4)(11)

Elementary Technical Mathematics

Expand sin(+45) and then simplify.

Trigonometry (MindTap Course List)

Exercise can be generalized as follows: If and the set has elements, then the number of elements of the pow...

Elements Of Modern Algebra

Evaluating a Binomial Coefficient In Exercises 85-88, evaluate the binomial coefficient using the formula (kn )...

Calculus of a Single Variable

Finding the Standard Equation of a Hyperbola In Exercises 41-48, find the standard form of the equation of the ...

Calculus: Early Transcendental Functions (MindTap Course List)

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Finite Mathematics for the Managerial, Life, and Social Sciences

For the right circular cylinder shown, r=4 cm and h=6 cm. Find the exact a lateral area. __ b volume. _____

Elementary Geometry for College Students

Using an Integrating Factor The differential equation (kx2+y2)dxkxydy=0 is not exact, but the integrating facto...

Multivariable Calculus

For y = f (u) and u = g(x), dydx=dydududx.

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Finding Points of Intersection In Exercises 15-18, find the points of intersection of the graphs of the equatio...

Calculus

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

Calculus (MindTap Course List)

0 1 ∞

Study Guide for Stewart's Multivariable Calculus, 8th

The data from exercise 1 follow. xi 1 2 3 4 5 yi 3 7 5 11 14 a. Use equation (14.23) to estimate the standard d...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Use a calculator to find each value to four decimal places. log0.00467

College Algebra (MindTap Course List)

Consider the following hypothesis test: H0: = 18 Ha: 18 A sample of 48 provided a sample mean x= 17 and a sa...

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

List the five sections typically found in a research article, and describe briefly what each should contain.

Research Methods for the Behavioral Sciences (MindTap Course List)

What is the goal of a hypothesis test?

Research Methods for the Behavioral Sciences (MindTap Course List)

A certain pen has been designed so that actual mean writing lifetime under controlled conditions (involving the...

Introduction To Statistics And Data Analysis

Three holes are drilled on a horizontal line in a housing. The center distance between the first hole and the s...

Mathematics For Machine Technology

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