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

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Chapter 7.6, Problem 37E
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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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Least-Cost Rule The production function for a company is given by

f ( x ,   y ) = 100 x 0.7 y 0.3 where x is the number of units of labor (at $50 per unit) and y is the number of units of capital (at $ 100 per unit). Management sets a production goal of 20,000 units.

(a) Find the numbers of units of labor and capital needed to meet the production goal while minimizing the cost.

(b) Show that the conditions of part (a) are met when

Marginal productivity of labor Marginal productivity of capital = Unit price of labor Unit price of capital

This proportion is called the Least-Cost Rule (or Equimarginal Rule).

(a)

To determine

To calculate: The number of units of labor and capital needed to meet a production goal while minimize the cost from the function f(x,y)=100x0.7y0.3.

Explanation

Given Information:

The production function for a manufacturer is given by f(x,y)=100x0.7y0.3x 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)=100x0.7y0.3

Minimize f(x,y)=50x+100y subject to constraint 100x0.7y0.3−20,000=0. now solve the constraint.

Fx(x,y,λ)=50x+100y−λ(100x0.7y0.3−20,000)=0

Since

Fx=50−70λx−0.3y0.3=0

And

Fy=100−30λx0.7y−0.7=0

And

Fλ=−(100x0.7y0

(b)

To determine

To proof: The condition of part (a) are met

marginal productivity of labourmarginal productivity of capital=unit price of labourunit price of capital from the function

f(x,y)=100x0.7y0.3.

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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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Surface area of a Single-Celled Organism The surface area 5 of a single-celled organism may be found by multipl...

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

Critical Thinking Consider two data sets. Set A:n=5;x=10 Set B:n=50;x=10 (a) Suppose the number 20 is included ...

Understanding Basic Statistics

A score that is 9 points above the mean corresponds to a z-score of z=1.50. What is the population standard dev...

Statistics for The Behavioral Sciences (MindTap Course List)

Sketch the graph of an example of a function f that satisfies all of the following conditions: limx0+f(x)=2,lim...

Single Variable Calculus

In a survey at a health club, 208 members indicated that they enjoy aerobic exercises, 145 indicated that they ...

Mathematical Excursions (MindTap Course List)

In problems 15-26, evaluate each expression. 26.

Mathematical Applications for the Management, Life, and Social Sciences

Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60...

Probability and Statistics for Engineering and the Sciences

Convert the expressions in Exercises 6584 to power form. 25x3

Finite Mathematics

For Problems 5-54, perform the following operations with real numbers. Objectives 3-6 581112

Intermediate Algebra

Find f. f(t) = 2t 3 sin t, f(0) = 5

Single Variable Calculus: Early Transcendentals

39. Assume that and are subgroups of the abelian group. Prove that the set of products is a subgroup of.

Elements Of Modern Algebra

20. You are the manager of Rally Rent-a-car. A customer, Sandy Furrow, has asked you for an estimate of charges...

Contemporary Mathematics for Business & Consumers

Prove that each of the following identities is true. cot2xsinx+cosx=cos2xsinxcos3x2sin4xsin2x

Trigonometry (MindTap Course List)

Divide: 100+25

Elementary Technical Mathematics

In Exercises 1 to 4, give the ratios in simplified form. a 15:24 c 2 m:150 cm 1 m = 100 cm b 2 ft:2 yd 1 yd = 3...

Elementary Geometry For College Students, 7e

Finding a Point of Intersection In Exercises 33-36, determine whether the lines intersect, and if so, find the ...

Calculus: Early Transcendental Functions (MindTap Course List)

Finding a Derivative In Exercises 29-40, Find the derivative of the algebraic function. f(x)=x4(12x+1)

Calculus of a Single Variable

In Exercises 9-12, let Z be the standard normal variable. Find z if z satisfies the given value. P(Zz)=.1469

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 20 to 22, apply the Midpoint Formula. A circle has its center at the point -2,3. If one endpoint o...

Elementary Geometry for College Students

Integration by Tables In Exercises 4956, use integration tables to find or evaluate the integral. 0/2x1+sinx2dx

Calculus

Using Properties of the GradientIn Exercises 6166, find the gradient of the function and the maximum value of t...

Multivariable Calculus

Finding the Volume of a Solid In Exercises 912, write and evaluate the definite integral that represents the vo...

Calculus (MindTap Course List)

The solution to y = y2 with y(1)=13 is: a) y=1x+2 b) y=lnx+13 c) y=12x216 d) y=12x16

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

Rewritten as an iterated integral in polar coordinates,

Study Guide for Stewart's Multivariable Calculus, 8th

Solve each of the following equations using the addition principle of equality. Check each answer. T12=34

Mathematics For Machine Technology

Find x. log0.25x=-1

College Algebra (MindTap Course List)

Explain the difference between a population characteristic and a statistic.

Introduction To Statistics And Data Analysis

Discuss the recursive nature of fractal described in each of the assigned exercises see Exercises 2-4.

Mathematics: A Practical Odyssey

Define a factorial research design, including the terms factor and level, and identity and describe factorial d...

Research Methods for the Behavioral Sciences (MindTap Course List)

A zoo has categorized its visitors into three categories: member, school, and general. The member category refe...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

What is the novelty effect, and how does it affect a studys external validity?

Research Methods for the Behavioral Sciences (MindTap Course List)

Motion Picture Industry The motion picture industry is a competitive business. More than 50 studios produce a t...

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

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