Cubic and Quaric Models 52 For noncommercial use Frank C. Wilson
USAA Producs (Cubic) Source: The Legacy of Membership: USAA 2006 Repor o Members s (since 1999) x USAA producs (millions) p 1999 0 14.7 2000 1 15.9 2001 2 16.8 2002 3 17.6 2003 4 18.4 2004 5 19.8 2005 6 21.8 2006 7 23.9 Scaer Plo and Model Graph Model Equaion px ( ) = 0.02904x 1972x + 1.279x+ 14.74 Verbal Descripion Based on daa from 1999 2006, he number of producs of Unied Services Auomobile Associaion may be modeled by px ( ) = 0.02904x 1972x + 1.279x+ 14.74 million producs where x is he number of years since 1999. (Source: Modeled from The Legacy of Membership: USAA 2006 Repor o Members) Quesions 1. According o he model, wha will be he number of producs in 2008? 2. Wha is he difference in he growh rae in producs beween 1999 and 2003 as compared o 2003 hrough 2006? (Explain how hey are differen. Don calculae he difference.) 3. According o he model, in wha year was he number of members increasing he mos slowly? 54 For noncommercial use Frank C. Wilson
USAA Membership (Cubic) Source: The Legacy of Membership: USAA 2006 Repor o Members s (since 1999) x USAA members (millions) m 1999 0 4.1 2000 1 4.4 2001 2 4.6 2002 3 4.8 2003 4 5.0 2004 5 5.2 2005 6 5.5 2006 7 5.9 Scaer Plo and Model Graph Model Equaion 2 wx ( ) = 0.1238x + 0.004762x+ 6.917 Verbal Descripion Based on daa from 1999 2006, he number of members of Unied Services Auomobile Associaion may be modeled by mx ( ) = 0.005808x 0.05206x + 0.3365x+ 4.102 million members where x is he number of years since 1999. (Source: Modeled from The Legacy of Membership: USAA 2006 Repor o Members) Quesions 1. According o he model, wha will be he number of members in 2008? 2. Where does he inflecion poin of he model occur and wha does i mean in his real world conex? 3. According o he model, in wha year was he number of members increasing he mos slowly? Frank C. Wilson For noncommercial use 55
Egg Producion (Cubic) Source: Saisical Absrac of he Unied Saes, 2006, Table 842 (Since 1990) E Egg Producion (billions) 0 68.1 5 74.8 7 77.5 8 79.8 9 82.9 10 84.7 11 86.1 12 87.3 13 87.5 14 89.1 Scaer Plo and Model Graph Model Equaion E( ) = 0.0125 + 0.266 + 0.199 + 68.1 Verbal Descripion Based on daa from 1990 2004, he number of eggs produced in he Unied Saes may be modeled by E() = 0.0125 + 0.266 + 0.199 + 68.1 billion eggs where is he number of years since 1990 (Source: Saisical Absrac of he Unied Saes, 2006, Table 842) Quesions 1. According o he model, how many eggs were produced in 1996? 2. According o he model, wha will be he maximum number of eggs produced beween 1990 and 2010? 3. Wha is he inflecion poin of he model and wha does i represen in is real world conex? Frank C. Wilson For noncommercial use 57
Per Capia Consumer Spending on Physician Services by (Cubic) Source: Saisical Absrac of he Unied Saes, 2006, Table 121 (Since 1990) P Per Capia Spending on Physician and Clinical Services (dollars) 0 619 5 813 8 914 9 954 10 1010 11 1085 12 1162 13 1249 Scaer Plo and Model Graph Model Equaion P = 0.3639 5.252 + 55.3 + 619.3 ( ) Verbal Descripion Based on daa from 1990 2003, he per capia spending on physician and clinical services in he Unied Saes may be modeled by P = 0.3639 5.252 + 55.3 + 619.3dollars () where is he number of years since 1999. (Source: Modeled from Saisical Absrac of he Unied Saes, 2006, Table 121) Quesions 1. According o he model, wha was he per capia spending in 1997? 2. Where does he inflecion poin of he model occur and wha does i mean in his real world conex? 3. Wha does he concaviy of he model graph ell us abou he rae a which per capia spending on physician and clinical services is increasing? 58 For noncommercial use Frank C. Wilson
Per Capia Spending on Healh Services and Supplies by (Cubic) Source: Saisical Absrac of he Unied Saes, 2006, Table 121 (Since 1990) T Toal Per Capia Spending on Healh Services and Supplies (dollars) 0 2633 5 3530 8 3962 9 4154 10 4389 11 4733 12 5115 13 5452 Scaer Plo and Model Graph Model Equaion T( ) = 1.676 24.69 + 257.3 + 2635 Verbal Descripion Based on daa from 1990 2003, he per capia spending on healh services and supplies in he Unied Saes may be modeled by T() = 1.676 24.69 + 257.3 + 2635 dollars where is he number of years since 1999. (Source: Modeled from Saisical Absrac of he Unied Saes, 2006, Table 121) Quesions 1. According o he model, wha was he per capia healh service and supply spending in 2002? 2. Where does he inflecion poin of he model occur and wha does i mean in his real world conex? 3. Wha does he concaviy of he model graph ell us abou he rae a which per capia spending on healh service and supply spending is increasing? Frank C. Wilson For noncommercial use 59
Pager Sales as a funcion of Cell Phone Sales (Cubic) Source: Saisical Absrac of he Unied Saes, 2006, Table 1003 (since 2000) p Pager Sales (millions of dollars C Cell Phone Sales (millions of dollars) 0 750 8995 1 790 8651 2 810 8106 3 729 9163 4 675 10538 Scaer Plo and Model Graph Model Equaion C( p) = 0.002745p + 6.173p 4633p+ 1,169,000 Verbal Descripion Based on daa from 2000 2004, cell phone sales may be modeled by C( p) = 0.002745p + 6.173p 4633p+ 1,169,000 million dollars where p is he pager sales (in millions of dollars) for he same year. (Source: Modeled from Saisical Absrac of he Unied Saes, 2006, Table 1003) Quesions 1. Does i seem reasonable ha cell phone and pager sales are relaed? Explain. 2. When pager sales are around 730 million dollars, he concaviy of he graph changes. Wha does his imply in he real world conex? 3. Does his model have any relaive exrema? Explain. Frank C. Wilson For noncommercial use 61
Male Life Expecancy as a funcion of Female Life Expecancy (Cubic) Source: Saisical Absrac of he Unied Saes, 2006, Table 96 Birh m Male Life Expecancy (years) f Female Life Expecancy (years) 1980 70.0 77.4 1982 70.8 78.1 1983 71.0 78.1 1984 71.1 78.2 1985 71.1 78.2 1986 71.2 78.2 1987 71.4 78.3 1988 71.4 78.3 1989 71.7 78.5 1990 71.8 78.8 1991 72.0 78.9 1992 72.3 79.1 1993 72.2 78.8 1994 72.3 79.0 1995 72.5 78.9 1996 73.0 79.0 1997 73.6 79.4 1998 73.8 79.5 1999 73.9 79.4 2000 74.3 79.7 2002 74.5 79.9 2003 74.8 80.1 Scaer Plo and Model Graph 2005 74.9 80.7 Model Equaion f m = 0.045645m 9.9292m + 720.34m 17350 2010 75.6 81.4 ( ) Verbal Descripion Based saisical daa, he life expecancy of an American female may be modeled by f m = 0.045645m 9.9292m + 720.34m 17350 years ( ) where m is he life expecancy of an American male (in years). (Source: Modeled from Saisical Absrac of he Unied Saes, 2006, Table 96) Quesions 1. The model is increasing everywhere. Wha does his mean in is real world conex? 2. A wha male life expecancy is he female life expecancy increasing mos slowly? 3. Wha does he concave up porion of he graph ell us abou he relaionship beween male and female life expecancies? 62 For noncommercial use Frank C. Wilson
Consumer Spending on Books (Cubic) Source: Saisical Absrac of he Unied Saes 2007, Table 1119 s (since 2004) x Consumer Spending on Books (million dollars) b 2004 0 49,146.6 2005 1 51,919.8 2006* 2 53,723.8 2007* 3 55,546.9 2008* 4 57,194.3 2009* 5 58,875.2 * projeced Scaer Plo and Model Graph Model Equaion bx ( ) = 41.01x 416.3x + 2999x+ 49180 Verbal Descripion Based on daa from 2004 2005 and projecions for 2006 2009, he amoun of money spen by consumers on books may be modeled by bx ( ) = 41.01x 416.3x + 2999x+ 49180 million dollars where x is he number of years since 2004. (Source: Modeled from Saisical Absrac of he Unied Saes 2007, Table 1119.) Quesions 1. Of wha pracical value is he model o book reailers? 2. On wha inerval(s) is he funcion increasing? 3. On wha inerval(s) is he graph of he funcion concave up? Frank C. Wilson For noncommercial use 69
Television Ses (Cubic) Source: Saisical Absrac of he Unied Saes 2007, Table 1111 s (since 1992) x Television Ses in Homes (millions) T 1992 0 192 1993 1 201 1994 2 211 1995 17 1996 4 223 1997 5 229 1998 6 235 1999 7 240 2000 8 245 2001 9 248 2002 10 254 2003 11 260 2004 12 268 2005 187 Scaer Plo and Model Graph Model Equaion T( x) = 0.06680x 1.276x + 12.46x+ 190.7 Verbal Descripion Based on daa from 1992 2005, he number of elevision ses in U.S. homes may be modeled by T( x) = 0.06680x 1.276x + 12.46x+ 190.7 million ses where x is he number of years since 1992. (Source: Saisical Absrac of he Unied Saes 2007, Table 1111.) Quesions 1. Based on he model, wha does he concaviy of he graph ell us abou he rae a which elevision ses in homes are increasing? 2. The populaion of he Unied Saes was 273 million in 1997 and 294 million in 2004. Which grew more rapidly beween 1997 and 2004, he number of TV ses or he number of people? 3. Assuming he populaion of he Unied Saes grows linearly, when will he number of elevisions in he Unied Saes equal he populaion? (The populaion of he Unied Saes was 273 million in 1997 and 294 million in 2004.) 70 For noncommercial use Frank C. Wilson