Explanation Given information: It is provided that the level of production is 1000 automobiles are maintained and there is new hiring of 150 workers by the rate of 10 per year and the automobile assembly plant has a production function is P = 10 x 0.3 y 0.7 where, P represents the total number of automobiles per year, x represents the total number of employees, and y represents the total daily operating budget. Formula used: The steps for solving the related rate problem are: Step1: List the changing and related quantities followed by restating the problem in rate of change of terms. Step 2: Rewrite the problem using for the changing quantities for their derivatives. Step 3: If possible draw the diagram of the changing quantities and find an equation relating the changing quantities. Step 4: Differentiate with respect to the equation (s) in order to get rate of change in relating quantity. Step 5: Now substitute the values of the provided quantity and their derivative into the derived equation and solve them to get answer. Calculation: Consider the steps of the formula: The changing quantities are the numbers P of automobiles produced per year, the number x of employees, and the daily operating budget y . Since, a production level of 1000 automobiles are maintained and there is new hiring of 150 workers by the rate of 10 per year. Thus, the rate of change in production is being constant at 1000 and the rate of change of x is 10 when x = 150 . Thus, d x d t = 10 when x = 150 and d P d t = 0 at P = 1000 Since, the rate at which daily operating budget be increasing is to be determined. Thus, the above sentence can be rephrased as: What is the rate of change of y . Thus, d y d t = ? Consider the relationship equation: P = 10 x 0.3 y 0.7 Substitute, 1000 for P , 150 for x in above equation to calculate the y as: 1000 = 10 ( 150 ) 0.3 y 0.7 y = ( 100 150 0.3 ) 1 0.7 y = 100 10 7 150 3 7 Differentiate the above equation of relationship with respect to t by using the product rule of differentiation which is: ( a b ) ′ = a ′ b + a b ′ Thus, substitute 10 x 0

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.5, Problem 29E

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To calculate: The rate at which daily operating budget be changing where the level of production is 1000 automobiles are maintained and there is new hiring of 150 workers by the rate of 10 per year and the automobile assembly plant has a production function is P=10x0.3y0.7 where, P represents the total number of automobiles per year, x represents the total number of employees, and y represents the total daily operating budget.

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Chapter 12.5, Problem 28E

Chapter 12.5, Problem 30E

Chapter 12 Solutions

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Solution Summary: The author explains how to calculate the rate at which daily operating budget decreasing is 2.40 per year.

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