Chapter 14.2, Problem 57E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Average ProductionThe Cobb-Douglas production function for an automobile manufacturer is f ( x , y ) = 100 x 0.6 y 0.4 , where x is the number of units of labor and y is the number of units of capital. Estimate the average production level when the number of units of labor x varies between 200 and 250 and the number of units of capital y varies between 300 and 325.

To determine

To calculate: The average value of production level and the production function for an automobile manufacturer is f(x,y)=100x0.6y0.4; where the number of units of labour x varies between 200 and 250, and the number of units of capital y varies between 300 and 325.

Explanation

Given:

The production function for an automobile manufacturer is f(x,y)=100x0.6y0.4,

The number of units of labour x varies between 200 and 250, and the number of units of capital y varies between 300 and 325.

Formula used:

The average value formula is:

AverageÂ value=1Aâˆ¬Rf(x,y)dA

Calculation:

First find the area of region R with the bounds 200â‰¤xâ‰¤250 and 300â‰¤yâ‰¤325.

Here, the area of region R is,

A=(50)(25)=1250

The average value of f(x,y) is provided by:

AverageÂ value=1Aâˆ¬Rf(x,y)dA

Now, put all the required values and get,

AverageÂ value=1Aâˆ¬Rf(x,y)dA=11250âˆ«200250âˆ«300325100x0

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