Warm-Up: Answer the following questions: Section 1.1: Modeling and Equation Solving 1. Which weighs more? A pound of iron or a pound of feathers? 2. What is alive and has only 1 foot? 3. When do giraffes have 8 feet? 4. How many eggs can you put in an empty basket? Problem-Solving Process ** Step 1: Understand the problem. a) Read the problem as stated. b) Understand the meaning of each math word." c) dentify the information you need to solve for. ** Step 2: Develop a mathematical model of the problem. a) DRAW A PCTURE b) ntroduce variables to represent quantities. c) Use the written statements to find an equation. d) Be creative. Just because do a problem one way, doesn't mean that is the only way. ** Step 3: Solve the mathematical model & support or confirm the solution. a) Solve using Algebra, numbers, or a graph. **Step 4: nterpret the solution in the problem setting. a) Decide whether the result makes sense.
. --.. ~ Table 1.1 The Minimum ~ Hourly Wage l Minimum Purchas ing Hourly Wage Power ill Year (MHW) 1996 Dollars 07srs 1955 4.39 1960 1.00.2s 5.30 1965 1.25 35 6.23 1970 1.60 ' 6.47 1975 2.10.~ 6.12 1980 3.10 1.00 5.90 1985 3.35.25 4.88 1990 3.80.Lt5 4.56 1995 45 4.25 4.38.q 2000 5.15 4.69 2005 5.15 0 4.15. EXAMPLE 1 Tracking the Minimum Wage The lumbers ill Table 1.1 show the growlll of the minimum hourly wage (MHW) from 1955 10 2005. The table also shows the MHW adjusted to the purchasi ng power of 1996 dollars (using the CP-U, the Consumer Price ndex for all Urban CUllsulllers). Answer the following questions lsing on ly the data in the tab le. (a) [n what tive-year period did the ac tu al MHW increase the most? (b) n wh at year did a worker earn ing the MHW enjoy the grelllesl purchasing power'? (e) A worker on minimum wage in 1980 was earning nearly twice as much as a wo rke r on minimuill w:jge in J970, and yet there was great pressure to raise the min imum wage ag<lin. Why'! a) ql5 - \Q80 b) glo 6) ~ Yl c) decreclsq b~ \l51 n q q~ ( nfl ailo n) EXAMPLE 3 Comparing Pizzas A pizzeria sells a recta ngular 18" by 24" pizza for the same price as it(targe ro u 1}d.~ / plzza (24" diameterj)jf both pizzas are of the same thickness, which option gives the most pizza for [he money? ~=LT(2. = T (\Z)l -:: 452
Year Jo Female 1980 3.8 1985 4.4 1990 5.S 1995 5.9 2000 6.7 growing over the years. g ~ 5 4 ~ 1 1~--4-~~r-~~ 'G 5 qa 'qs QO \QBo 0 iqoo. \L\-ftJX -'285.28 ode] this growth graphically and lise the EXAMPLE 6 Solving an Equation Algebraically Find all real numbers x for which 6.\:3 = 1 x 2 + lox. tox3 -\\X l -\OX ~O X ( ~X 2 - l:x- --10) ::- 0 )( (3x. + z)(x- ')) ~() ~
EXAMPLE 7 Solve the equation x 2 = 10-4x. _~:\ G1 rq ph 'ieros J', -= X1. :: to '4x Solving an Equation: Comparing Methods XL-lO t4)( :: 0 )( 2 -r1 X - to = 0 1.7 EXAMPLE 8 Applying the Problem-Solving Process The engineers at an auto manufacturer pay students $0.08 per mile plus $25 per day to road test their new vehicles. (a) How much did the auto manufacturer pay Sally to drive 440 miles in one day? (b) John earned $93 test-driving <1 new car in one day. How far did he drive'? j :: _08 m t 25 d d ) ~"" b) 93 ::.08 m t Z~.08 (14aJ T25 () -:: $ ~O 10 ty\ :: 950
EXAMPLE 9 Seeing Grapher Failure Look at the graph of.r = 3/(2x - x-intercept? 5) on a graphing calculator. [s there an NO ~ --/NT EXAMPLE 10 Not Seeing Hidden Behavior Solve graphically: x J - l.lx 2-65.4x + 229.5 = O. EX l)qnd tht window ai 3 X-nt to ieq.,
n Exercises 1-10, match the numerical model to the corresponding graphical model (a-j) and algebraic model (k-f), AP --t 401 by [-. 7] (e) [-3, 18) by [-2, 321 (d) 1-1. 7J by [-4, 40J (e) [- 1,7) by [-4, 40J (0 1-1, 16J by [-, 9J (g) [-5, 30] by (-5, 100) (h) s. x 4 8 12 14 18 24 ~. Y 20 72 156 210 142 600 } x 3 4 5 6 7 8 '---'" v 8 15 24 35 48 63 10' 1 x 4 7 12 19 28 39 )' 2 3 4 5 6 / V V [-2. 14] by [-4. 36) [-, 6J by [-2, 201 (a) (b) M [-191 by (-2, 601 (i) (k)'y = x 2 + X (m)\' = (x + )(x ) (0) \' = loo - 2x (q) y = 2x (s) y =2x + 3 [-5,401 by [-10, 650J (j) () y =40 - x 2 (n)y= vx=-3 (p) y =3x - 2 (r) y =x 1 +2 1-3 (t)y= 2
Year Female 1954 32.3 1959 35. J 1964 36.9 1969 41. 1974 42.8 1979 47.7 1984 50.1 1989 54.9 1994 56.2 1999 58.S 2004 57.4 83.5 82.3 80.9 81. 1 77.9 76.5 73.2 74.5 7'2.6 74.0 71.9 12. (a) According to the numerical model, what has been the trend in males joi ni ng tile work force since 1954? (b) n what 5-year interval did the percentage of men who were employed change the most? Q) ~C~CASln9 b) q7q - q~t
~. ~..-...- --...... '"' Table 1.7 U.S. Air' Travel' Passengers Year (millions) Yenr 1991 452.3 1998 1992 475.1 1999 1993 488.5 :2000 1994 528.8 2001 1995 547.8 2002 J996 58J.2 2003 1997 594.7 2004 ~ -..."'... - 0''".. -~-..- ~'2... "'.' Passengers (millions) 612.9 636.0 666.1 622.1 612.9 646.3 697.8 (a) Graph a scatter plot of the data. Let x be the number of years since 1991. (b) From 1991 to 2000 the data seem to follow a linear model. Use the 1991 and 2000 points to find an equat.ion of the line and supelimpose the line on the scatter plot. (e) According to the linear model, in what year did [he number of passengers seem destined to reach 900 million? (d) What happened to disrupt the linear model') ~SD b) [STAT}~[CALCr!f\ re3 ~25 boo 51),. j:: 23.10}; +452'-5 S50 625 500 tj15 Q50 0 c) qbovt :2010 d) q- 7 9 q 10 12 (3 J~
)' 1.400,000 1,260,000 1,120.000 980.000 840,(X)() 700,000 560.000 420,000 280,(X)O 140,000 ~~~~~~~~~X -1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 After Peter Ueberroth's resignation as baseball commissioner in 1988 and his successor's untimely death in 1989~ the team owners broke free of previous restrictions and began an era of competitive spending on player salaries. dentify where the 1990 salaries appear in the graph and explain how you can spot them. ~~r to