## Question (a)

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ALGEBRA

Thecost of fees and tuition for public colleges can be found throughsimple summation of all the costs from the year 2007 to 2010 ending.

Thesimple summation can be expressed as the total sum and is given by

Equation (1) Total sum= w + x + y + z

Where

W represents costs in year 2007

X represents costs in year 2008

Y represents costs in year 2009

Z represents costs in year 2010

Fromthe bar graph above, the following values of fees and tuition forpublic colleges has been collated together with the correspondingyears

2007 ending = 5,836

2008 ending = 6,185

2009 ending = 6,585

2010 ending = 7,020

w=5,836, x= 6,185, y= 6,585, and z= 7,020

Usingequation (1) above

Totalsum= w + x + y + z

Substitutingthe values of w, x, y and z respectively

=5,836+ 6,185 + 6,585+ 7,020

=25,626

Thetotal cost of fees and tuition at public colleges is 25,626 for thefour-year period ending 2010.

Question(b)

1. Given a model an= 395n + 5419 to describe the cost of fees and tuition at public colleges

Then

an= 395n + 5419where n=1

Substituting

a1=395* 1 + 5419

=5,814

an=395n + 5419 where n=2

Substituting

a2=395* 2 + 5419

=6,209

an=395n + 5419 where n=3

Substituting

a3=395* 3 + 5419

=6,604

an=395n + 5419where n=4

Substituting

a4=395* 4 + 5419

=6,999

Theformula for Sn represents the sum total offees and tuition costs from 2007 to year 2010 ending.

Sncan be given by the following equation (2)

Sn = S1+ S2+ S3 + S4

Where

S1,S2, S3 and S4 represents the valuesof an calculated from the model an=395n + 5419 given above.

ThereforeS can be substituted as below

S1= 5,814, S2 = 6,209, S3= 6,604and S4= 6,999

Andusing equation (2) above

Sn = S1+ S2+ S3 +S4

SubstitutingS in the above equation

Sn = 5,814 + 6,209 + 6,604 + 6,999

=25,626

Thetotal fees and tuition cost at public colleges for the four-yearperiod is 25,626.

1. In comparison, the values for individual years vary from the actual values when the model is used.

Table1: Comparison of Actual fees and tuition Cost with estimated costsusing the model

 YEAR ENDING ACTUAL VALUES MODEL VALUES DIFFERENCE 2007 5,836 5,814 +22 2008 6,185 6,209 -24 2009 6,585 6,604 -19 2010 7,020 6,999 +21 TOTAL 25,626 25,626 0

Fromthe values on the table presented above, the actual cost of fees andtuition for the year ending 2007 and 2010 are more than the costspredicted by the model by 22 and 21 respectively. At the same time,the cost of fees and tuition for the years ending 2008 and 2009 areless compared with the predicted values from the model by 24 and 19respectively.

Further,the total sum of the costs of fees from the year 2007 to 2010 isequal for both the actual value obtained from the bar graph and thevalue from the model (25, 626 and 25,626 respectively).

Fromthese two methods, there is a convergence point at the total sum forfour years. It shows that the model can be used to predict the exactlump sum amount an individual can pay for the four years. Otherwise,because of small differences between the individual years from 2007to 2010 as 22, 24, 19, and 21 respectively, it will be misleading topredict the exact amount and individual should pay each year for thefour years running using the model.

Reference

Zariski, O.,&amp Samuel, P. (2013).&nbspCommutativealgebra&nbsp(Vol.2). Springer Science &amp Business Media