Quadratic Function Models

Transcription

1 Quadraic Funcion Models Frank C. Wilson For noncommercial use 19

2 Tenpin Bowling Membership as a funcion of Bowling Esablishmens (Quadraic) Source: Saisical Absrac of he Unied Saes, 006, Table 134 (Since 1990) b Tenpin Bowling Esablishmens M Bowling Membership (Thousands) Model Equaion M ( b) = b 6.61b Based on daa from , he number of enpin bowling memberships may be modeled by M ( b) = b 6.61b where b is he number of enpin bowling esablishmens (Source: Saisical Absrac of he Unied Saes, 006, Table 134). 1. Based on he raw daa, wha would you hink a pracical domain for his model would be?. According o he model, how many bowling memberships will here be when here are 6800 enpin bowling esablishmens? 3. According o he model, how many bowling esablishmens will here be when here are 4000 enpin bowling memberships? Frank C. Wilson For noncommercial use 1

3 Make I Real Book of Mahemaical Models Projeced Naional Healh Expendiures: Public vs. Privae (Quadraic) Source: Saisical Absrac of he Unied Saes, 006, Table 118 p Privae Expendiures Insurance and Ou-of-Pocke (billion dollars) G Public Expendiures Federal and Sae (billion dollars) proj proj proj proj proj proj Model Equaion proj G( p) = p 0.71 p 7.01 Based on daa from and projecions for , he amoun of public money spen on healh care may be modeled by G( p) = p 0.71 p 7.01 billion dollars where p is he amoun of privae money (ou-of-pocke and privae insurance) spen on healh care (in billion dollars) (Source: Saisical Absrac of he Unied Saes, 006, Table 118). 1. Of wha pracical value is his mahemaical model?. Public healh expendiures are funded by he public hrough axes. A wha rae are public expendiures expeced o be increasing when he amoun of privae expendiures reaches 1500 billion dollars? 3.Wha does he concaviy of his model indicaes abou healh spending? Frank C. Wilson For noncommercial use 7

4 Prescripions as a funcion of Medicare Enrollees (Quadraic) Source: Saisical Absrac of he Unied Saes, 006; Tables 16, 13 s (since 1995) Medicare Enrollees (Millions) m Prescripions (millions) p Model Equaion p( m) = 43.74m m Based on daa from , he number of prescripions (in millions) may be modeled by p( m) = 43.74m m where m is he number of Medicare enrollees (in millions) (Source: Modeled from Saisical Absrac of he Unied Saes, 006; Tables 16, 13). 1. Wha does he concaviy of he graph ell us abou his siuaion?. According o he model, how many prescripions will be wrien when here are 43 million Medicare employees? 3. How many Medicare Enrollees were here when 500 million prescripions were wrien? Frank C. Wilson For noncommercial use 9

5 Children Under 5 Sleeping Under Insecicide Treaed Nes Chad (Quadraic) Source: hp:// Proporion of Children Under 5 Sleeping under Insecicide Treaed Nes (ITN) C 000 ( = 0) % 001 ( = 1) 6.7% 00 ( = ) 10.0% 005 Goal 60.0% Model Equaion C = using = 0,1, 5 ( ) ( ) C = using = 0,1, Based on daa from and a goal for 005, he proporion of children under 5 in Chad sleeping under insecicide reaed nes may be modeled by C( ) = cases where is he number of years since 000 (Source: Based on daa from , he proporion of children under 5 in Chad sleeping under insecicide reaed nes may be modeled by C( ) = cases where is he number of years since 000 (Source: 1. Wha does he concaviy of each model ell us abou he rae a which he percenage of children sleeping under he ITNs in Chad is changing?. Of wha value would hese models be o a healh worker combaing malaria in Chad? 3. Wha is he difference beween he average raes of change of he wo models on he inerval [0,5]? Frank C. Wilson For noncommercial use 31

6 Source: Populaion in Equaorial Guinea (Quadraic) s (since 1990) Populaion (housands) P Model Equaion P( ) = Based on daa from , he populaion of Equaorial Guinea may be modeled by P( ) = people where is he number of years since 1990 (Source: 1. In 1990, here were 5,55 clinical malaria cases repored in Equaorial Guinea. If he same percenage of he populaion conraced malaria in 010 as in 1990, how many people would be infeced wih he disease in 010?. Wha is he pracical meaning of he verical inercep of he model? 3. According o he model, when will he populaion firs reach 1,000,000? Frank C. Wilson For noncommercial use 33

7 Hospial Beds as a funcion of Hospials (Quadraic) Source: Healh, Unied Saes, 006, p. 364 Hospials h Beds b ,156 1,465, ,965 1,364, ,87 1,317, ,649 1,13, ,634 1,01, ,539 1,177, ,467 1,163, ,374 1,18, ,91 1,080, ,01 1,061, ,097 1,035, ,01 1,01, , , , , , , , , , , , ,768 Model Equaion bh ( ) = h 157h+ 4, 931, 000 Based on daa from , he number of hospial beds as a funcion of he number of hospials may be modeled by bh ( ) = h 157h+ 4, 931, 000 where h is he number of hospials (Source: Healh, Unied Saes, 006, p. 364 ). 1. From he graph of he model, i appears ha he hospial bed funcion is concave up and increasing. Neverheless, from he daa we see ha he number of hospial beds has been decreasing every year. Do he graph and he raw daa conflic? Explain.. According o he model, wha is he minimum number of hospial beds ha here will ever be and how many hospials will here be a ha ime? 3. If we modeled hospials as a funcion of hospial beds, wha would be he concaviy of he resulan graph? Explain. 36 For noncommercial use Frank C. Wilson

8 Consumer Spending on Books: Adul Trade (Quadraic) Source: Saisical Absrac of he Unied Saes 007, Table 1119 s (since 004) x Adul Trade Book Spending (million dollars) b , , * 15, * 3 16, * 4 16, * 5 16,985.4 * projeced Model Equaion bx ( ) = 31.15x x Based on daa from and projecions for , he amoun of money spen by consumers on books classified as adul rade may be modeled by bx ( ) = 31.15x x million dollars where x is he number of years since 004. (Source: Modeled from Saisical Absrac of he Unied Saes 007, Table 1119.) 1. According o he model, i wha year will he amoun of money spend by consumers on adul rade books reach a maximum?. Wha is he pracical meaning of he verex of he model graph in his conex? 3. Wha does he concaviy of he graph mean in his conex? 38 For noncommercial use Frank C. Wilson

9 Arizona Virual Academy Enrollmen (Quadraic) Source: Arizona Virual Academy Fac Shee s (since ) x Enrolled Sudens s Model Equaion sx ( ) = 141.5x x Based on daa from hrough , he number of sudens enrolled in he Arizona Virual Academy may be modeled by sx ( ) = 141.5x x sudens where x is he number of years since (Source: Modeled from Arizona Virual Academy Fac Shee) 1. According o he model, wha will be he enrollmen in ?. If you were an execuive a he Arizona Virual Academy, of wha value would his model be o you? 3. Do you hink he model will be valid for 00 01? Explain. 4 For noncommercial use Frank C. Wilson

10 Aborion Rae (Quadraic) Source: Saisical Absrac of he Unied Saes 007, Table 96 s (since 1985) x Aborions (Per 1000 Live Birhs) a Model Equaion ax ( ) = 0.167x 8.339x Based on daa from , he aborion rae in he Unied Saes may be modeled by ax ( ) = 0.167x 8.339x aborions per 1000 live birhs where x is he number of years since (Source: Modeled from Saisical Absrac of he Unied Saes 007, Table 96.) 1. Wha is he verex of he model and wha does i mean in is real world conex?. Aborion is one of he mos divisive issues in American sociey. How could aciviss on boh sides of he issue benefi from he aborion rae model? 3. According o he model, when will he aborion rae sop decreasing and wha will be he aborion rae a ha ime? 46 For noncommercial use Frank C. Wilson