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

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Chapter 7.6, Problem 35E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Problem 6SWU
Problem 7SWU
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Problem 9SWU
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Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
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Problem 29E
Problem 30E
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Problem 33E
Problem 34E
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Problem 39E
Problem 40E
Problem 41E
Problem 42E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Production The production function for a manufacturer is given by

f ( x ,   y ) =   100 x 0.25 y 0.75 where x is the number of units of labor (at $48 per unit) and y is the number of units of capital (at $36 per unit). The total cost for labor and capital cannot exceed $100,000.

(a) Find the maximum production level for this manufacturer.

(b) Find the marginal productivity of money.

(c) Use the marginal productivity of money to find the maximum number of units that can be produced when $125,000 is available for labor and capital.

(d) Use the marginal productivity of money to find the maximum number of units that can be produced when $350,000 is available for labor and capital.

(a)

To determine

To calculate: The maximum production level of the manufacturer from the function

f(x,y)=100x0.25y0.75.

Explanation

Given Information:

The production function for a manufacturer is given by f(x,y)=100x0.25y0.75x is the no of unit of labor and y is the no of unit of capital

Formula used:

Method of Lagrange multipliers,

If the function f(x,y) contains a maximum or minimum subject to the constraint g(x,y)=0 then the maximum or minimum can occur at one of the critical numbers of the function F is,

F(x,y,λ)=f(x,y)−λg(x,y) where, λ is a Lagrange multiplier.

Steps to determine the minimum or maximum of the function f.

1. Solve the system of equations,

Fx(x,y,λ)=0Fy(x,y,λ)=0 Fλ(x,y,λ)=0

Calculation:

Consider the function,

f(x,y)=100x0.25y0.75

Maximize f(x,y)=100x0.25y0.75 subject to constraint 48x+36y=100000. now solve the

constraint.

Fx(x,y,λ)=100x0.25y0.75−λ(48x+36y−100000)

Since

Fx=25x−0.75y0.75−48λ=0

And

Fy=75x0.25y−0.25−36λ=0

And

Fλ=−(48x+36y−100000)=0

Using F=xFy

F,xλ=25x−

(b)

To determine

To calculate: The marginal productivity of money from the function f(x,y)=100x0.25y0.75.

(c)

To determine

To calculate: The maximum number of units that can be produce when $125000 is available for labor and capital r from the function f(x,y)=100x0.25y0.75.

(d)

To determine

To calculate: The maximum number of units that can be produce when $350,000 is available for labor and capital from the function f(x,y)=100x0.25y0.75.

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Chapter 7.6,
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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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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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P-56RECh-7 P-57RECh-7 P-58RECh-7 P-59RECh-7 P-60RECh-7 P-61RECh-7 P-62RECh-7 P-63RECh-7 P-64RECh-7 P-65RECh-7 P-66RECh-7 P-67RECh-7 P-68RECh-7 P-69RECh-7 P-70RECh-7 P-71RECh-7 P-72RECh-7 P-73RECh-7 P-74RECh-7 P-75RECh-7 P-76RECh-7 P-77RECh-7 P-78RECh-7 P-79RECh-7 P-80RECh-7 P-81RECh-7 P-82RECh-7 P-83RECh-7 P-84RECh-7 P-85RECh-7 P-86RECh-7 P-87RECh-7 P-88RECh-7 P-89RECh-7 P-90RECh-7 P-91RECh-7 P-92RECh-7 P-93RECh-7 P-94RECh-7 P-95RECh-7 P-96RECh-7 P-97RECh-7 P-98RECh-7 P-99RECh-7 P-100RECh-7 P-101RECh-7 P-102RECh-7 P-103RECh-7 P-104RECh-7 P-1TYSCh-7 P-2TYSCh-7 P-3TYSCh-7 P-4TYSCh-7 P-5TYSCh-7 P-6TYSCh-7 P-7TYSCh-7 P-8TYSCh-7 P-9TYSCh-7 P-10TYSCh-7 P-11TYSCh-7 P-12TYSCh-7 P-13TYSCh-7 P-14TYSCh-7 P-15TYSCh-7 P-16TYSCh-7 P-17TYSCh-7 P-18TYSCh-7 P-19TYSCh-7 P-20TYSCh-7 P-21TYSCh-7 P-22TYS

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

a. What is a function? b. What is the domain of a function? The range of a function? c. What is an independent ...

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

Use the method of cylindrical shells to find the volume generated by rotation the region bounded by the curves ...

Calculus (MindTap Course List)

18 . 18. For the following scores, find the value of each expression: X 3 5 0 2 X X2 X+1 (X+1)

Statistics for The Behavioral Sciences (MindTap Course List)

Statistical Literacy What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is ...

Understanding Basic Statistics

Determine the infinite limit. 37. limx2xcscx

Single Variable Calculus

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

Intermediate Algebra

In Exercises 17 to 24, let U = {2. 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14,...

Mathematical Excursions (MindTap Course List)

In Problems 47-50, rationalize the denominator of each fraction and simplify. 50.

Mathematical Applications for the Management, Life, and Social Sciences

An engineering construction firm is currently working on power plants at three different sites. Let Ai denote t...

Probability and Statistics for Engineering and the Sciences

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

Finite Mathematics

(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate ...

Single Variable Calculus: Early Transcendentals

a. How many years will it take $5,000 invested at 8 simple interest to double to $10,000? b. How long will it t...

Contemporary Mathematics for Business & Consumers

True or false Label each of the following statement as either true or false. Let and be integers, not both zer...

Elements Of Modern Algebra

Find the weight of the display model shown in Illustration 3. The model is made of pine. Pine weighs 31.2 lb/ft...

Elementary Technical Mathematics

Solve the equation. Check your solutions when necessary 4(1x2)+4x5=0

Trigonometry (MindTap Course List)

In Exercise 5 to 9, use a compass and a straightedge to complete the constructions. Given: Triangle ABC Constru...

Elementary Geometry For College Students, 7e

Minimum Distance In Exercises 1316, find the point on the graph of the function that is closest to the given po...

Calculus: Early Transcendental Functions (MindTap Course List)

Evaluating a Definite Integral In Exercises 9-36, evaluate the definite integral. Use a graphing utility to ver...

Calculus of a Single Variable

In exercises 27 to 33, complete a formal proof. If two consecutive angles of a quadrilateral are supplementary,...

Elementary Geometry for College Students

The problems in Exercises 6182 correspond to those in Exercises 2344, Section 2.1 Use the results of your previ...

Finite Mathematics for the Managerial, Life, and Social Sciences

Pendulum The period T of a pendulum of length L is T=(2L)/g where a is die acceleration due to gravity. A pendu...

Calculus

True or False: The function y = f(x) graphed at the right could be a solution to y = x + y.

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

MatchingIn Exercises 7176, match the equation (written in terms of cylindrical or spherical coordinates) with i...

Multivariable Calculus

Vertical Motion In Exercises 97 and 98, use the position function s(t)=4.9t2+v0t+s0 for free-falling objects. A...

Calculus (MindTap Course List)

Use the Taylor polynomial of degree 2 for centered at 1 to estimate . 1.250 1.255 1.265 1.270

Study Guide for Stewart's Multivariable Calculus, 8th

Multiply the following terms as indicated. (x)(x2)

Mathematics For Machine Technology

An automobile mileage study tested the following hypotheses. Hypothesis Conclusion H0: 25 mpg Manufacturers c...

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

A box contains five slips of paper, marked 1, 1, 1, 10, and 25. The winner of a contest selects two slips of pa...

Introduction To Statistics And Data Analysis

Identify the statistical techniques that are appropriate for each nonexperimental, quasi-experimental, and deve...

Research Methods for the Behavioral Sciences (MindTap Course List)

Use the division algorithm and synthetic division to express the polynomial function P(x)=3x32x26x4 in the form...

College Algebra (MindTap Course List)

A person is interested in constructing a portfolio. Two stocks are being considered. Letx = percent return for ...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

According to the following articles, the rock group Pink Floyd reduced its volume to protect Venices buildings....

Mathematics: A Practical Odyssey

Select a subject and use a full-text database such as PsycARTICLES to locate an empirical journal article that ...

Research Methods for the Behavioral Sciences (MindTap Course List)

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