Explanation Given: The table shows the net profits (in millions of dollars) for Microsoft from 2007 through 2014. Year 2007 2008 2009 2010 Net profit (in millions) 14,065 17,681 14,569 18,760 Year 2011 2012 2013 2014 Net profit (in millions) 23,150 23,171 22,453 22,074 Calculation: Consider the points ( 2007 , 14065 ) , ( 2008 , 17681 ) , ( 2009 , 14569 ) and ( 2010 , 18760 ) . As the x -values are large, so use the translation z = x − 2000 , Then the points becomes, ( 7 , 14065 ) , ( 8 , 17681 ) , ( 9 , 14569 ) and ( 10 , 18760 ) The cubic polynomial function be, p ( z ) = a 0 + a 1 z + a 2 z 2 + a 3 z 3 Substituting z = 7 , 8 , 9 and 10 into p ( z ) and equating the results to the respective y -values produces the system of linear equations in the variables a 0 , a 1 , a 2 and a 3 as shown below. p ( 7 ) = a 0 + a 1 ( 7 ) + a 2 ( 7 ) 2 + a 3 ( 7 ) 3 = a 0 + 7 a 1 + 49 a 2 + 343 a 3 = 14065 p ( 8 ) = a 0 + a 1 ( 8 ) + a 2 ( 8 ) 2 + a 3 ( 8 ) 3 = a 0 + 8 a 1 + 64 a 2 + 512 a 3 = 17681 p ( 9 ) = a 0 + a 1 ( 9 ) + a 2 ( 9 ) 2 + a 3 ( 9 ) 3 = a 0 + 9 a 1 + 81 a 2 + 729 a 3 = 14569 p ( 10 ) = a 0 + a 1 ( 10 ) + a 2 ( 10 ) 2 + a 3 ( 10 ) 3 = a 0 + 10 a 1 + 100 a 2 + 1000 a 3 = 18760 The augmented matrix is, [ 1 7 49 343 14065 1 8 64 512 17681 1 9 81 729 14569 1 10 100 1000 18760 ] Solving the system by using Gauss elimination with back substitution: − R 1 + R 2 → R 2 , − R 1 + R 3 → To determine b) To calculate: The solution of the system and also determine does the solution produce reasonable model for determining net profits after 2010..

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

8th Edition

ISBN: 9781337604925

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 1.3, Problem 19E

To determine

a)

To calculate:

The system of equations to fit the data for the years 2007, 2008, 2009, and 2010 to a cubic model.

To determine

b)

To calculate:

The solution of the system and also determine does the solution produce reasonable model for determining net profits after 2010..

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Chapter 1.3, Problem 18E

Chapter 1.3, Problem 20E

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Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term

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a) To calculate: The system of equations to fit the data for the years 2007, 2008, 2009, and 2010 to a cubic model.

Solution Summary: The author calculates the system of equations to fit the data for the years 2007, 2008, 2009, and 2010 to a cubic model.