Project II - Modeling Quadratic Functions The table below shows the operating re
ID: 3116280 • Letter: P
Question
Project II - Modeling Quadratic Functions The table below shows the operating revenue and expenses of 25 U.S. Schedule Service Passenge Statistics) er Airlines Financial Report in 2015. (Source: Bureau of Transportation Year Operating Revenue Operating (millions of dollars) Expenses (millions of dollars) 2010 2011 2012 2013 2014 2015 130.517.70 148,048.00 150,467.00 150,285.70 154,676.60 138,827.00 153,293.90 156,470.30 161.596.70 169,276.50 168,874.20 140,881.00 1. Find the quadratic model that is the best fit for the operating revenue, with x equal to number of years after 2000 (First report the unrounded function and then round to three decimal places). Label this function R(x). Find the quadratic model that is the best fit for the operating expenses, with x equal to number of years after 2000 (First report the unrounded function then round to three decimal places). Label this function E(x). 2.Explanation / Answer
1.Let the operating revenue function be R(x) = ax2+bx+c, where x is the number of years since 2000 and a,b,c are real numbers. In 2010, when x = 10, we have R(2010) = a*102+b*10+c = 100a+10b +c = 138827…(1). In 2011, when x = 11, we have R(2011) = a*112+b*11+c = 121a+11b +c = 153293.90…(2) and in 2012, when x = 12, we haveR(2012)= a*122+b*12+c =144a+12b+c =156470.30…(3). The augmented matrix of this linear systemof equations is A =
100
10
1
138827
121
11
1
153293.90
144
12
1
156470.30
The RREF of A is
1
0
0
-22581/4
0
1
0
2660343/20
0
0
1
-1253639/2
Thus, a = -22581/4, b = 2660343/20 and c = -1253639/2
Then R(x) = -(22581/4)x2+(2660343/20)x -1253639/2 = -5645.25 x2+133017.15 x -626819.5
2. Let the operating expenses function be E(x) = dx2+ex+f, where x is the number of years since 2000 and d,e,f are real numbers. In 2010, when x = 10, we have E(2010) = d*102+e*10+f = 100d+10e +f = 130517.70…(1). In 2011, when x = 11, we have E(2011) = d*112+e*11+f = 121d+11e+f = 148048…(2) and in 2012, when x = 12, we have E(2012)= d*122+e*12+f =144d+12e+f =150467…(3). The augmented matrix of this linear systemof equations is B =
100
10
1
130517.70
121
11
1
148048
144
12
1
150467
The RREF of B is
1
0
0
-151113/20
0
1
0
3523979/20
0
0
1
-4379534/5
Thus, d = -151113/20,e = 3523979/20 and f =-4379534/5
Then E(x) = -(151113/20)x2+(3523979/20)x -4379534/5 = -7555.65x2+176198.95x -875906.80.
100
10
1
138827
121
11
1
153293.90
144
12
1
156470.30
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