Quantitative Skills & Reason
Quantitative Skills & Reason MATH 1001
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This 9 page Class Notes was uploaded by Carolyn Farrell on Monday October 12, 2015. The Class Notes belongs to MATH 1001 at Gordon College taught by Marwan Zabdawi in Fall. Since its upload, it has received 70 views. For similar materials see /class/222094/math-1001-gordon-college in Mathematics (M) at Gordon College.
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Date Created: 10/12/15
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slum m ewadnucfmmmexmnplel m 174m cmbewnmnas x 7 x7 17 mewmucrmmmaxmpxez gx 72x14x3 canbewntlenas gx 72x7115 Nmethanhepanhnlamexample 1 apmswwm wahvenexV rlj mdavemcal axxs uf symmetxy Th2 panhaan example 2 apens dwwnwazdthhvenex m 5 mamch axxsafsymmekyxl swam Farm uh Quzdn Funtl Wm Gnyh Pznhuh us Vn lua A Thegaphuf m 15 ns may is a parabala um has vertex m k and a vemcal ms The parabola opens upward if a gt 0 or downward if a lt 0 If a gt 0 the vertex Vh k is the lowest point on the parabola and the function f has a minimum value f h k If a lt 0 the vertex Vh k is the highest point on the parabola and the function f has a maximum value f h k Quadratic Models and Equations Quadratic Population Models Pt P0 bt atz Here we denote the independent variable by ttime instead of x and the constant c by P0 because substitution of t 0 yields P0 P0 We refer to P0 as the initial population Example 3 Suppose that the future population of Stockton City tyears after January 1 2000 is described in thousands by the quadratic model Pt 110 4t 007t2 a What is the population of Stockton City on January 1 2007 b In what month of what calendar year will the population of Stockton City reach 180 thousand Solution a We only need to substitute t 7 in the population function Pt and calculate P7 110 47 00772 14143 thousand b We need to find the value oftwhen Pt 180 That is we need to solve the equation 1104t007t2 180 139 Equamn 1 an example cf 5 quadm c equa nn whnnh we can eme m a variety nfmys Firs by gnphing bath sxdes mime equaunn gure 3 y e 11042oo722 yze xmmcuon 2 musssu v1ao gure 3 The lme 12 20 and me parabnla y e no 4r a 07 are shnwnmthe calcmatnrwmdnw rm 5 r 20 rum Sy s 300 Tn eme equaunn t we nd me chnnrdmate mime rmereeeum perm m the rst quadrant The negeuve snluunn mhe mtersecnnn pmnt m the secnnd quadrant wnu d be m the past Fxgure 3 mdmazes that we have already used the mtersecnnnr ndmg feature w Incmethepnmt 14 047 180 Hence t 14 047 years 14 years a 047 lljmnnths 14 years o 55 manth Thus smmn Cxty shnmd reach epepmeum man mausand 14 years a 55 manth a erlmuary 1 2000 Hence snmeume dunnglanuary 2m Altsm valy by usmg me Quarm39e anmlh Thequadramcequazmn ax2bxc o 110 4 2 Aac 2a has snluunns r To use the quadratic formula we first write equation 139 in the form 007t2 4t 70 0 ii Here a 007 b 4 c 770 giving t 4 r 42 4007 70 2007 4615396 14046954 or 71189811 The negative solution would be in the past So we only accept the positive solution I 14047 years 14 years 056 month The Position Function Model of a Palticle Moving Vertically If an object is projected straight upward at time t 0 from a point y0 feet above ground with an initial velocity voftsec then its height above ground after t seconds is given by yt l6t2 vot yo Example 4 A projectile is red vertically upward from a height of 600 feet above the ground with an initial velocity of 803 ftsec a Write a quadratic model for its height ht in feet above the ground after t seconds b During what time interval will the projectile be more than 5000 feet above the ground c How long will the projectile be in ight sum a Gwznyn6 and Vn8U3 jseq m 7161 2m rm We medm swam values uftfaxwhchthz 1mmfo gt mun rm mm 3 45mm 6nn gt mun 0 We salve m mzquahtyO bygaphmgmgm 0 mm parabala y 7161 mm mm mum y sum m39hecalcmntmwmdaw 7m s x 5 7n and iznnn s y mun me gure t gue 4 mm 1mm 2mm Mme y mm m W11 uf hagmsuuu rm Th2 mm 5 um mun rm shave m gmwm m Nb um pm 5 shave m mm m Th2 parabnla m hammul km 1 Uangthz 1mxsechmfmdmgfemxe arm mum we nd the appmxxmnle mursemanpmn39stabef 63 sec mdt439 52 Hence 39heprmeculexs abWeSEIEIEI feelwhzn 6352clttlt 439 sec Th2 pmjechle m be mmgmmm nsmgmo n This cmespandstathz xrm39zxcepmnm he mgmn gmu Uangthz mm mzemfeamre fth mm W abum r m 9 sec EXERCISES l The population in thousands for Alpha City t years after January 1 2004 is modeled by the quadratic function Pt 03t2 6t 80 In what month of what year does Alpha City s population reach twice its initial 112004 population The population in thousands for Beta City t years after January 1 2005 is modeled by the quadratic function Pt 07t2 l2t 200 How long will it take Beta City s population to reach 350 thousand The population in thousands for Gamma City t years after January 1 2002 is modeled by the quadratic function Pt 15t2 2lt 300 How long will it take Gamma City s population to reach 500 thousand The population in thousands for Delta City t years after January 1 2003 is modeled by the quadratic function Pt 05t2 7t 90 In what month of what year does Alpha City s population reach twice its initial 1 12003 population The population in thousands for Omega City t years after January 1 2002 is modeled by the quadratic function Pt 025t2 5t 100 In what month of what year does Alpha City s population reach 200 thousand A ball is thrown straight up from ground zero with an initial velocity of 48 feet per second Find the maximum height attained by the ball and the time it takes for the ball to return to ground zero From the top of a 48 feet tall building a ball is thrown straight up with an initial velocity of 32 feet per second Find the maximum height attained by the ball and the time it takes for the ball to hit the ground A ball is thrown straight up from the top of a 160 feet tall building with an initial velocity of 48 feet per second The ball soon falls to the ground at the base of the building How long does the ball remain in the air A ball is dropped from the top of a 960 feet tall building How long does it take the ball to hit the ground Joshua drops a rock into a well in which the water surface is 300 feet below ground level How long does it take the rock to hit the water surface Fitting Quadratic Models to Data Find the quadratic model Pt P0 bt atz with t 0 for the earliest year given in the data that best ts the population census data in Problems 11 7 16 In each case calculate the average error of this optimal model and use the model to predict the population in the year 2007 11 Iowa City IA tyears 1970 1980 1990 2000 Ppeople 46850 50508 59735 62220 Arizona tyears 2000 2001 2002 2003 2004 Pthous 5131 5320 5473 5581 5744 Florida tyears 1970 1980 1990 2000 2004 Pthous 6789 9746 12938 15982 17346 Georgia I ears I 1970 1980 1990 2000 2004 Pthous 4590 5463 6478 8186 8825 Nevada t ears I 1970 1980 1990 2000 2004 Pthous 489 801 1202 1998 2315 US tyears 1970 1980 1990 2000 2004 Pgmillions 203 227 249 281 294 Answers to Exercises 908994P N February 2013 8 years 143 days 6 years 183 days February 2011 May 2014 36 feet 3 sec 64 feet 3 sec 5 sec 775 sec 433 sec Pt 46963 807t 95t2 319pe0ple 63000 people Pt 5139 176t 69t2 14thous 6030th0us Pt 6777 298t 037t2 44thous 18300th0us Pt 4610 59t193t2 59thous 9400th0us Pt 502 14t116t2 31thous 2600th0us Pt 204 2t 002t2 14thous 300thous Note Answers to questions 1116 are approximate values References 1 Elementary Mathematical Modeling Functions anal Graphs Mary Ellen Davis amp C Henry Edwards Prentice Hall 2001 Intermediate Algebra R David Gustafson amp Peter D Frisk Thomson BrooksCole 2005 WWWfairusorg WWW gpecorg WWW johnson county com
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