Chapter 2 Modeling with Linear Functions

Transcription

1 Chapter Modeling with Linear Functions Homework.1. a. b. c. When half of Americans send in their tax returns, p equals 50. When p equals 50, t is approximately 10. Therefore, when half of Americans sent in their tax returns, the year was approximately 005. d. When p equals 80, t equals approximately 1, which represents the year Therefore, it is estimated that the 80% tax return percentage will be achieved by the year 01. This model predicts the goal will not be achieved. 8. a. c. The elevation 8850 meters is represented by E Locating this on the linear model, note that B is approximately 160. Therefore, the boiling point of water is 160 F at the peak of Mount Everest. Interpolation was used, owing to our examination of predicted data in between points. d. According to the linear model, when B 98.6, the elevation is approximately 19 thousand meters. Therefore, at an elevation of 19,000 meters, boiling water feels lukewarm. Extrapolation was used, owing to the examination of predicted data outside of the data points. e. Answers may vary. Keep in mind that the higher the boiling point (lower elevation), the less time it takes for an egg to cook. Likewise, the lower the boiling point (higher elevation), the more time it takes for an egg to cook. The linear model should be increasing. e. The goal will not be reached by 011, according to the estimate.. Responses for this exercise are the same for parts (c) and (d) of Exercise. Responses for this exercise are different for parts (e) and (f) of Exercise. Explanations may vary. 6. a. b. b. c. An estimated amount of carbon emissions in 000 (t 50) is.5 billion tons according to the linear model. d. The actual amount of carbon emissions in 000 was 6.. This is 1. billion tons less than predicted by the model. This comparison suggests that carbon emissions may have begun to increase less rapidly. Students curves may vary.

2 10. a. 6. The equation of the line is n.0t The equation of the line is c 0.015w b. c. According to the model, the estimated percentage of people living below the poverty level in 00 (t 1) was approximately 9.%. d. e. The estimated value from part (c) was 9.%. The actual value for 00 was 1.%. The error in prediction is 9.% 1.%.%. The linear model breaks down after the year 000, so the predicted value is much too low. 1. a. To find the t-intercept, estimate what the temperature will be when w 0. This estimate is approximately (1, 0). This means that if the wind speed is 10 mph and the temperature is 1 degrees Fahrenheit, it will feel like it is 0 degrees Fahrenheit. b. To find the w-intercept, estimate what the windchill will be when t 0. This estimate is approximately (0, 16). This means that if the wind speed is 10 mph and the temperature is 0 degrees Fahrenheit, it will feel like it is 16 degrees Fahrenheit below We usually have more faith in a result obtained by interpolation. Explanations may vary. In this model, m needs to be greater (steeper slope) than it was in the original model. The value of b should decrease. 1. Use (1, 18) and (10, 5) to write the equation of the line. First, find the slope m So, y 1.x + b. Substitute 1 for x and 18 for y since the line contains (1, 18) and then solve for b. y mx+ b 18 1.() 1 + b b 19. b The equation of the line is y 1.x (Your equation may be slightly different if you chose different points.) Use the graphing calculator to check your results. 1. Check the scatter plot obtained from the graphing calculator. 16. Answers may vary. Homework.. The equation of the line is n 1.8t The equation of the line is n 8.6t 10.. Student C made the best choice of points. The lines obtained by students A and B will have a slope that is steeper than the actual slope. 5

3 a. p 0.69t The slope will be the same but the p-intercepts will be different. Though the data is the same (hence the same slope), using a different number for the year shifts the graph, and therefore the value of p. b. Use the points (0,.) and (0,.0) to find the equation of the line..0. m So, L 0.165t + b. Substitute 0 for t and. for L since the line contains (0,.) and then solve for b. L mt+ b ( 0) + b. 0 + b. b The linear equation that describes the data is L 0.165t+.. (Your equation may be slightly different if you chose different points.) c. 1 m So, p 0.0a + b. Substitute for a and for p since the line contains (, ) and then solve for b. p ma+ b 0.0( ) + b b 1.8 b The equation of the line is p 0.0a 1.8. (Your equation may be slightly different if you chose different points.) d. The equation of the line is p 0.1a e.. a. The two graphs are very similar. c. 0. a. b. Use the points (, ) and (6, 1) to find the equation of the line. b. Use the points (16,.) and (88,.9) to find the equation of the line..9. m So, r 0.05t + b. Substitute 16 for t and. for r since the line contains (16,.) and then solve for b. 6

4 c. r mt+ b ( ) b b 8.1 b The equation is r 0.05t (Your equation may be slightly different if you chose different points.) d. i. Aggravated assaults continued to decline even though the economy did poorly, which supports the theory that crime and the economy are not related. ii. Burglaries declined until 000, then rose slightly after 000, indicating that the poor economy may have increased the number of burglaries. iii. All types of crime declined until 000, then remained fairly steady. This indicates that the poor economy may have caused crime to stop declining. 8. Answers may vary. The line comes close to the data points.. a. The equation of the line is r 0.080s b. The equation of the line is r 0.09s c. The women s equation has the larger r- coordinate (1.66). This tells you that the women s regression line is above the men s line when s 0. d. The women s model has a slightly larger slope. This indicates the line rises more quickly in comparison to the men s line. e. The women s line begins above the men s line and also rises faster than the men s line, so the two lines can never intersect in quadrant 1. Women, in general, have faster stride rates, regardless of speed. 6. a. A linear model works well, since a line can be drawn that comes close to the data points. The regression equation is A 15.t (Your equation may be slightly different if you used data points to form a model.) b. No, the last three data points don t fit a linear model. c. No, the last three data points don t fit a linear model. Homework.. f ( ) 6( ) f ( ) 6( ) f( a ) 6( a ) 6a 18 6a 8. g () ( ) 5() (9) 5() g( ) (( ) ) 5( ) () 5( ) 8 ( 10) ( ) h( ) 5( ) ( a) 9a h( a) 5( a) + 15a+ 16. g( 1) ( 1) + ( 1) f ( ) ( ) h( 9). f( a) ( a) 1a. a a 1a f ( ) ( ) 6a

5 6. f( a ) ( a ) a + 16 a f( a h) ( a h) a+ h 0. To find x when f( x ) 0, substitute 0 for f ( x) and solve for x: 0 x + x x. To find x when f( x ) f ( x) and solve for x: x + 9x + 1 9x 1 9x 5 5 x 9, substitute. To find x when f ( x) a+, substitute a + for f ( x) and solve for x: a+ x+ a 5 x a 5 x 5 a x 6. f (1.8) 5.95(1.8) To find x when f( x ).06, substitute.06 for f ( x) and solve for x: x x x x There is a mistake in the use of the order of for operations principle. Parentheses and exponents must be dealt with first before the other operations. Correctly done, the procedure is as follows: g( 5) ( 5) 5. a. f (5) (5) f () () f(5) f() 1 1 This value and the slope are the same. b. f () () f (6) (6) f() f(6) 19 1 This value and the slope are the same. c. f () () f () () f() f() 1 9 This value and the slope are the same. d. f( a+ 1) m( a+ 1) + b ma + m + b f( a) m( a) + b ma + b f ( a+ 1) f( a) ma+ m+ b ma b m This value and the slope are the same. This indicates that for every increment of one in a linear function, the next value will have a y value increased by the slope value.. f ( ) 0 6. When f ( x ), x is. 8. Since the line contains the point (0, ), f (0) Since the line contains the point, 6, 11 f Since the line contains the point (, 1), x when f(x) or y 1. 8

6 5. Since the line contains the point (.5,.5), x.5 when f(x) or y Since the line contains the point,, x when f(x) or y The range is < f( x) <. 60. Since the function contains the point (,), x when g(x) or y. 6. gx ( ) 6. Since the function contains the point (, 1), x when h(x) or y f ( x) or y x+ To find the y-intercept, set x 0 and solve for y y ( 0) + 0+ The y-intercept is (0, 1 ). To find the x-intercept, set y 0 and solve for x. 1 0 x + 1 x 1 x The x-intercept is (, 0). 66. hx ( ) f ( x) or y x+ To find the y-intercept, set x 0 and solve for y. y 0 ( ) + y 0+ The y-intercept is (0, ). To find the x-intercept, set y 0 and solve for x. 0 x + x x 1 1 The x-intercept is (, 0). 0. f ( x) or y x To find the y-intercept, set x 0 and solve for y. y ( 0) 0 The y-intercept is (0, 0). To find the x-intercept, set y 0 and solve for x. 0 x 0 x The x-intercept is (0, 0).. f ( x) or y is the equation of a horizontal line. The y-intercept of the line will be (0, ). There is no x-intercept f ( x) or y.9x+.58 To find the y-intercept, set x 0 and solve for y. y.9(0) +.58 y.58 The y-intercept is (0,.58). To find the x-intercept, set y 0 and solve for x. 0.9x x.58 x 8.6 The x-intercept is (8.6, 0). 8. a. Your table may be different if you chose different x values. b. x f ( x) c. For each input-output pair, the output is 1 less than times the input. 5

7 80. a. f () t 0.t b. ( ) ( ) f When t 110, f This means that in the year , the percentage of births outside marriage will be about 95.6%. This result is different than Exercise 9. c. 0.t t t.9 When f, t. This means that the percentage of births outside marriage will hit % in the year This is the same as the result obtained in Exercise 9. d t t t 116 In the year , all births will be outside marriage. This is the same as the result from Exercise 9. e. Find f, when t f ( ) 0.( ) This means that in the year 199, the percentage of births outside marriage was 1.%. Since the actual percentage was.%, the error is 1.%.% 0.%. This is nearly the same result obtained in Exercise a. The regression equation is f( t) 1.1t (Your equation may be slightly different if you used two data points to form a model.) The model fits the data well. b. f (15) 1.1(15) The model predicts that in the year , Ford s U.S. market share will be about 1.0%. c. To find t when f () t is 15, substitute f () t with 15 and solve for t: t t 1. t Ford s market share would have been at 15% in the year d. f () t s. To find the t-intercept, set s equal to 0 and solve for t t t 9.56 t 5.6 The t-intercept is (5.6, 0). The model predicts that in the year , Ford will have no percentage of the U.S. market share. e. To find the s-intercept, set t equal to 0 and solve for s. s 9.56 The s-intercept is (0, 9.56). The model estimates that in the year 1995, Ford controlled about 9.6% of the U.S. market share. 8. a. Since t is in years since 1980, we need t L (1) 0.16(1) Life expectancy for people born in the year 011 will be approximately 9 years. b. To find the birth year at which life expectancy is 81 years, we must find t at Lt () 81. To find this, substitute 81 for L and solve for t t t 5. t Since t is the number of years from 1980, is the birth year when the life expectancy will be 81. c. Answers may vary. One answer is as follows. For a person born in 1980, t 0. Find f (0). f (0). 0

8 86. a. Therefore, the estimated life expectancy of an American born in the year 1980 is approximately. years. Generally, a person s current life expectancy tends to be greater than it was at the time of birth. d. To find the t-intercept, set Lt () 0and solve for t t t 61 t This says that in the year , life expectancy at birth was zero. Model breakdown has occurred. e. Answers may vary. The graph should show that human life expectancy has increased more rapidly in the recent past than earlier in human history. b. The regression equation is P 0.011t + 1.8, where the population is in millions. (Your equation may be slightly different if you used two data points to form a model.) Graphing the line with the scattergram, we see that the model fits the data fairly well. c. Since , find f(50). f ( 50) 0.011( 50) According to the model, Detroit s population was 819,000 in the year 000. The actual count in 000 was 951,0. Detroit s efforts to increase its population count resulted in a 951,0 819,000 1,000 person increase. d. The data in the table can be converted into the points, (0, 1,0,9) and (50, 951,0). Use the points to find the slope. 951, 0 1, 0, 9 m So, g 60t + b. Substitute 50 for t and 951,0 for g since the line contains (50, 951,0) and then solve for b. g mt+ b 951, 0 60( 50) + b 951, 0 8, b 1,,0 b The linear model is g t t+. ( ) 60 1,,0 e. ( ) ( ) f , or 608,000 g ( 60) 60( 60) + 1,, 0 60, ,, 0 8, 50 Answers will vary. If Detroit s efforts continue, the better prediction will be g ( 60) 8, 50, because this model reflects recent attempts to increase the population. This means that in 010, Detroit s population will be about 85, a. Two of the data points are (50, ) and (90, 15). To find the equation for g, find the slope using these points m So, g ( F).F + b. To solve for b, substitute 50 for F and for g since the line contains (50, )..( 50) + b 15 + b 1 b g F F. The equation for g is ( ). 1 1

9 b. g ( ).( ) When the temperature is F, crickets will chirp approximately 1 times per minute. c. 100.F 1.F 6. F When the temperature is 6 F, crickets will chirp approximately 100 times per minute. d. When crickets are not chirping, g( F ) 0. We need to find F, when g( F ) 0. 0.F 1 1.F 0 F When the temperature is 0 F or below, crickets will not chirp. 90. a. f( a) 0.a 1.91 b. f (0) 0.(0) , or about.9% c. To find a when f ( a ) is %, substitute for f ( a) and solve for a: 0.a a a 9.9, or about age 50 d. To find the a-intercept, substitute 0 for f ( a) and solve for a. 0 0.a a.1 a The model states that at approximately age, the percentage of Americans with diabetes is 0. Model breakdown has occurred. e. Answers will vary. Perhaps fewer people over 0 are checked for diabetes. 9. From the given information, we have the data points (1955, 1.) and (000, 8.9). Use these points to find the slope of the equation for f m So f ( t) 0.16t+ b. By substituting 000 for t and 8.9 for f () t, we can calculate b (000) + b b 5.1 b Thus the function is f( t) 0.16t 5.1. To find the number of words in 010, we substitute t with 010. f( t) 0.16(010) 5.1 f( t) f() t In 010 there will be about 10.6 million words in the federal tax code. 9. From the given information, we have the data points (195, 10) and (00, ). We can use these points to find the slope of the linear model m So f ( t) 0.8t+ b. To find b, substitute 00 for t and for f () t and solve for b. 0.8(00) + b 961+ b 98 b This makes the linear model f( t) 0.8t 98. To find when % will occur, substitute % for f () t and solve for t. 0.8t t 010. t This makes the time approximately 010 or a. From the given information, we have the data points (1990, 56.) and (005, 50.1). We can use these to find the slope of a linear model: m This makes the model f ( t) 0.0t+ b. To find b, we substitute 50.1 for f () t and 005 for t and solve for b:

10 (005) + b b b f( t) 0.0t To find when the percentage will be %, we substitute for f () t and solve for t. 0.0t t 01.5 t The percentage will be % in approximately 01 or 01. b. To find the percentage in 010, substitute t 010 and solve for f () t. f( t) 0.0(010) f( t) f() t 8.0 The percentage will be about 8% in We are given two data points, (, 06) and (6, 11). We can now calculate the slope of the linear model m 5.5, which 6 yields f ( n) 5.5n+ b, where n is the number of people in the family. We can now substitute for n and 06 for f ( n) to get b () + b b 9 b f( n) 5.5n+ 9 To find the income limit for a family of, substitute for n and solve for f ( n ). f () 5.5() + 9 f () f () a. From the given information, we have the data points (0, 1) and (5, 0). Use these points to find the slope of the equation for f m So, f () t.t+ b. Since the line contains (0, 1), substitute 0 for t and 1 for f ( t ) and solve for b. b. ( ) b 1 b The linear model is f () t.t+ 1. c. Since it takes 5 minutes to eat the ice cream, the domain is 0 t 5. Since there are 1 ounces of ice cream, the range is 0 f t 1. ( ) 10. Answers may vary. The four steps are: 1. Create a scattergram of the data to determine whether there is a nonvertical line that comes close to the data points. If so, choose two points (not necessarily data points) that you can use to find the equation of a linear model.. Find an equation of your model.. Verify your equation by checking that the graph of your model contains the two chosen points and comes close to all of the data points.. Use the equation of your model to make estimates, make predictions, and draw conclusions. Homework thousand 59.0 thousand rate thousand.95 thousand The average rate of change is.95 thousand spectators per year.. 55,5 51,05 1,5 rate The average rate of change is about 80 people per year rate The average rate of change is 0.% per year. 8. 1, 60, 51 96, 86, 98 rate 1,899 1

11 The average rate of change is $1,899 per bedroom. 10. a. t (hr) d (mi) b. There is a linear relationship because the train is moving at a constant speed. The slope is 0 miles per hour, so the distance increases 0 miles for each hour of travel. 1. a. There is an approximate linear relationship. The slope is $0. billion per year, which is the annual increase in sales of skin care products. b. The s-intercept of the model is $. billion. This is the sales of skin care products in 00, the base year. c. st ( ) 0.t+. d. ($billion) ($billion) (year) + ($billion) year ($billion) ($billion) + ($billion) ($billion) ($billion) e. To predict this value, substitute for s() t and solve for t. 0.t t 10 t Retail sales of skin care products will be approximately $ billion in a. The slope is 0.9. The percentage of workers in a union is decreasing by 0.9% per year. b. f( t) 0.9t+ 1 c. To find this quantity, we substitute f () t with 11 and solve for t t t 6.9 t The model predicts that 11% of workers will be in a union in 011. ( ) d. 010 is 6 years away from 00. Thus t 6. We substitute 6 for t and solve for f () t. f( t) 0.9t+ 1 f( t) 0.9(6) + 1 f( t) Approximately 11.% of workers will be in a union in 010. e. To find the t-intercept of the model, set f () t to 0 and solve for t t t.8 t The model predicts that no U.S. worker will be in a union in the year Model breakdown has likely occurred. 16. a. The slope of the graph of h is dollars per credit hour. This takes into account the tuition and the general fee per credit hour. For each additional credit hour, the cost increases by 60 dollars. b. hc ( ) 60c+ 15 c. cost($) cost($) (credit hour)+cost($) credit hour cost($)cost($)+cost($) cost($)cost($) d. To find h(9), substitute 9 for c in the equation we found and compute. h(9) 60(9) + 15 h(9) h(9) 555 This indicates that for 9 credit hours, the

12 student s total bill will be $555. e. To find c when hc () is 5, substitute 5 for hc () and solve for c. 5 60c c 1 c This indicates that if the student pays $5 for one semester, the student is taking 1 credit hours. 18. a. The slope is 0.00 atm/ft. This means that for every foot you descend, the pressure increases by 0.00 atm. b. f( d) 0.00d + 1 c. atm atm (ft) + atm ft atm atm + atm atm atm d. Twice the pressure at sea level would be atm. We must find the depth where this pressure occurs. To do this, substitute for f ( d) and solve for d. 0.00d d.0 d The depth at which the pressure is atm is ft. e. To find the pressure at 19 feet, substitute 19 for d and compute. f (19) 0.00(19) + 1 f (19) f (19) 59.8 The pressure at Crater Lake s greatest depth is about 60 atm. 0. Select t as the time in years since 001. The linear function may then be modeled as f( t) 1t+. To find t when f () t is 650, substitute 650 for f () t and solve for t t t 9.8 t 5 There will be approximately 650,000 patent applications in The customer paid for n gallons of gas, and 1 halfgallon of milk, with a total of 9. spent. Since we know the prices for these items, we can set up an equation to find n n n 1 n. The slope of this model is 0., indicating that the percentage of births occurring outside of marriage in the United States increases by 0. per year. 6. The slope of this model is 0.5, indicating that the percentage of adult Americans who smoke decreases by 0.5 per year. 8. The slope of this model is., indicating that for every degree Fahrenheit increase in temperature, the number of cricket chirps increases by. per minute. 0. a. The regression equation is f( t) 0.05t Your equation may differ slightly if you used two data points to form a model. b. The slope of the model is 0.05, indicating that the population of Nevada increases by 0.05 million per year. c. To predict the population in 01, we substitute t into the model, and calculate. f () 0.05() f () f ().85 f ().9 In 01 Nevada s population will be about.9 million. d. To find the t-intercept of the model, set f () t to 0 and solve for t t t 11. t The model indicates that Nevada s population was zero in This is false. Model breakdown has occurred.

13 e. To find this quantity, substitute. for f () t and solve for t t t.6 t Nevada s population will reach. million in about 010. ( ). a. The data cannot be modeled well with a linear model. The data points are scattered and there is no good line that would include most of them. b rate The average rate of change of injuries from mobile rides is zero. This is not a good description of the scattered data. c. The data can be modeled fairly well. The regression equation that models this data is It ( ) 0.51t.6. Your equation may differ slightly if you chose two data points to form a model. d. The number of injuries from inflatable rides increases by about 51 (or 0.51 thousand) per year. e. To find the year, substitute 8000/ for I() t and solve for t t t 0. t Since t is calculated in years since 1990, the year will be , or approximately 011. On average, 8000 injuries is 8000/ per state. c rate(0 100) rate(0 80) rate(0 60) The slope varies only slightly in these intervals, and agrees very closely with the slope found in part (b). d. To find the relative humidity, substitute.6 for f ( p) and solve for p p p 5. p The relative humidity is approximately 6%. 6. a. Accelerating from 50 mph to 0 mph in One Minute Time t (hours) d (in miles) 1: (0) 0 1: (1) 50 :00 50() 100 : : (1) 11 : () 1 5: () 11 b. 8. Answers may vary. One example is as follows.. a. The regression equation for the data is f( p) 0.10 p Your answer may differ slightly if you used two data points to model the data. b. The slope of the model is The slope tells us that for every increase of one in the relative humidity there will be a 0.10 degree increase in the heat index. 6

14 0. Answers may vary. One example is as follows.. Answers may vary. Chapter Review Exercises 1. f () () (9) 0. f ( ) ( ) (9) 0. () + 5 g() () h () ha ( + ) 10( a+ ) 10a 0 10a 6. 6 x + 6 x + 9 x 9 9 x or x. x + x + x 1 1 ( x ) x or x a+ x+ a+ x+ a+ x a+ x a a + x or x + 9. Since the graph includes the point (, 0), f() Since the graph includes the point (0, 1), f(0) Since the graph includes the point (,.5), f( ) Since the graph includes the point (, ), f( ). 1. Since the graph includes the point (, 0), f() Since the graph includes the point (, 1), f() The domain of f is 5 x The range of f is y. 1. Since x is 0 when f(x) 1, f(0) Since x is 1 when f(x), f() 1.

15 19. When f(x) 0, x. 0. When f(x), x f ( x) or y x+ To find the x-intercept, set y 0 and solve for x. 0 x + x x To find the y-intercept, set x 0 and solve for y. y ( 0) + y 0+ y 5. solve for y;.56(0) 9.1y y 8.5 y 8. The "New line" has an increased slope since it is steeper than the "Model." The y-intercept is lower in the "New line" equation as it intersects the y-axis at a lower point than the "Model.". f ( x) or y is the equation of a horizontal. line. The y-intercept of the line will be y. There is no x-intercept. f ( x) or y x+ To find the y-intercept, set x 0 and solve for y. y ( 0 ) + y 0+ y To find the x-intercept, set y 0 and solve for x. 0 x + x x x x. To find the x-intercept, set y equal to zero and solve for x;.56x 9.1(0) x 8.5 x 0.5 To find the y-intercept, set x equal to zero and 8 6. a. The student s car has 1 gallons of gas in the tank when no hours have been driven, so when t 0, b 1. Since the car uses 1.8 gallons of gas per hour, the slope is 1.8. An equation for f is f t t+. ( ) b. The slope of f is 1.8. This means that the amount of gasoline decreases at a constant rate of 1.8 gallons per hour of driving. c. To find the A-intercept of f, let t 0. A f ( t) 1.8( 0) The A-intercept is (0, 1). This represents the amount of gas in a full tank before any time has been spent driving (t 0). At that time, the car has 1 gallons of gas in the tank. d. gallons gallons gallons hour hour gallons gallons gallons This unit analysis shows that this model uses the correct units. e. To find the t-intercept of f, let f(t) t t 1 t t. The t-intercept is (., 0). This means that

16 the student can drive for. hours before running out of gas. f. Since the car can be driven for. hours before running out of gas, the domain is 0 t.. Since the gas tank has between 0 and 1 gallons of gas, the range is 0 f 1.. a. Since the average household income increases by $96 per year, the slope of g is a. The equation for s is s 0.9t +. To find the year when sales at duty-free shops worldwide reach $1 billion, we can substitute 1 for s and solve for t t t t According to the model in , sales at duty-free shops worldwide will reach $1 billion. b. In 00, t 0 and the average household income was $,9. Therefore, b,9. Since the slope is 96, the equation for g is g(t) 96t +,9. c. To find g(6), substitute 6 for t. g (6), (6), , 09 For this model g(6) 8,09, is the average U.S. personal income in d. Substitute 0,000 for g(t) and solve for t. 0, t +, 9, 06 96t t or t The average household income will be $0,000 in the year The rate of change in the number of times President George W. Bush used the word economy in the State of the Union addresses per year is given by the difference in the usage of the word divided by the difference in years. change in usage 16 m change in years If we let t be the number of years after 1980, and s the sales at duty-free shops worldwide we can make a linear model of the given information. To start, we can use the two points (0, ) and (5, 5) to find the slop of the model: 5 m So s 0.9t + b. To solve for b substitute 0 for t and for s since the line contains (0, ). 0.9(0) + b b 9 The data points lie close to the line that passes through (9, 1.) and (1,.5). To find the equation for f, start by finding the slope using these points m So, f(t) 1.t + b. To solve for b, substitute 9 for t and 1. for f(t) since the line contains (9, 1.) (9) + b b 0. b The equation for f is f(t) 1.t Graphing the equation on the same graph as the scattergram, we see that the equation fits the data very well. b. The slope of the model is 1.. This means that for every year between 1999 and 00, the standard mileage rate increases by 1. cents per mile. c. The M-intercept is given by letting t equal zero and solving for M. M 1.(0) + 0. M 0.

17 1. a. This means that during 1990 the standard mileage rate was 0. cents per mile. d. To predict when the standard mileage rate will be 5 cents per mile, substitute 5 for M and solve for t. 5 1.t t 0 t The model predicts that the standard mileage rate will be about 5 cents per mile in e. 01 corresponds to t. M 1.() + 0. M M.6 The model predicts that the standard mileage rate in 01 will be.6 cents per mile. If someone will drive 1,500 miles in 01 their deduction will be given by the standard mileage rate times the distance they traveled; 1,500(.6)590,50 This person will be able to deduct $5,90.50 since the standard mileage rate is in cents per mile. f. The estimate of 1. is calculated when each standard mileage rate is weighted by the number of months it was used. (.5) + 9(1) The data points lie close to the line that passes through (10, 8.) and (1, 8.9). To find the equation for f, start by finding the slope using these points m So, f(t).5t + b. To solve for b, 50 substitute 10 for t and 8. for f(t) since the line contains (10, 8.). 8..5(10) + b b b The equation for f is f(t).5t. Graphing the equation on the same graph as the scattergram, we see that the equation fits the data very well. b. The slope is.5. This means that for every year that passes between 1998 to 00 lobbying expenditures by Fortune 500 technology companies increased by $.5 million. c. To find the t-intercept, set f(t) equal to zero and solve for t. 0.5x.5x 1.98 x According to the model, in 1990 there were $1.98 million in lobbying expenditures by Fortune 500 technology companies. d. f (0).5(0) The model predicts that in , lobbying expenditures by Fortune 500 technology companies will reach $6.6 million. e. 0.5t.5t 8 t According to the model, in , there was $0 million in lobbying expenditures by Fortune 500 technology companies. f..5t 9.5t t

18 Chapter Test According to the model, in , lobbying expenditures by Fortune 500 technology companies will reach $9 million. 1. Since the line includes the point (, ), f ( ).. Since the line includes the point (, 0), f() 0.. Since the line includes the point (0, 1), f(0) Since the line includes the point 5,, 8 f ( 5) or approximately.. 5. Since the line includes the point ( 6, ), x 6 when f (x). 6. Since the line includes the point (, ), x when f (x).. Since the line includes the point (, 0), x when f (x) Since the line includes the point (.5, 0.5), x.5 when f (x) The domain of f is all the x-coordinates of the points in the graph. In this case f has a domain of 6 x The range of f is all the y-coordinates of the points in the graph. In this case f has a range of y To find f ( ), substitute for x. f ( ) ( ) To find f ( a 5), substitute a 5 for x. f( a 5) ( a 5) + a + 0+ a To find x when f ( x ), substitute a for f ( x ). x + 5 x 5 x 5 x 1. To find x when f ( x) a, substitute for f ( x ). a x+ a x a a+ x or x 15. To find the x-intercept, let f (x) 0 and solve for x. 0 x x x x The x-intercept is,0. To find the y-intercept, let x 0 and solve for y or f(x) since y f(x). y (0) y The y-intercept is (0, ). 16. To find the x-intercept, let g(x) 0 and solve for x. 0 x 0 x x 0 The x-intercept is (0, 0). To find the y-intercept, let x 0 and solve for y or g(x) since y g(x). y (0) y 0 The y-intercept is (0, 0). 1. To find the x-intercept, let k(x) 0 and solve for x.

19 0 1 x x x x The x-intercept is (, 0). To find the y-intercept, let x 0 and solve for y or k(x) since y k(x). y 1 (0) 8 y 8 The y-intercept is (0, 8). 18. a. Use the first three steps of the modeling process to find the equation. Begin by creating a scattergram using the data in the table. A line close to the data points passes through (16, 56.8) and (0, 81.8). Use those points to find the slope of the line m So f (a) 6.5a + b. To solve for b, substitute 16 for a and 68.8 for f (a) since the line contains (16, 56.8) (16) + b b. b The equation for f is f (a) 6.5a.. Your equation may be slightly different if you chose other points. The graph of the model fits the data well. b. The slope of the model is 6.5. This means that the percentage of teens with driver s licenses increases by about 6.5 every year after their 16 th birthday. 5 c a.. 6.5a a According to the model, 0% of year olds have a drivers license. While this might be true, it implies that some percentage of 8 though 15 year olds do have a driver s license, something that probably doesn t happen. d. Let a 1 and solve for f(a). f (1) 6.5(1) This means that percent of 1-year-old adults have a driver s license. e. Let f(a) 100 and solve for a a a a This means that 100 percent of -year-old adults have a driver s license. 19. a. Use the first three steps of the modeling process to find the equation. Begin by creating a scattergram using the data in the table. A line close to the data points passes through (65, 1.) and (90, 11.1). Use those points to find the slope of the line m So f (t) 0.96t + b. To solve for b, substitute 65 for t and 1. for f (t) since the line contains (65, 1.) (65) + b b.5 b b.5

20 The equation for f is f (t) 0.96t.5. (Your equation may be slightly different if you chose other points. The linear regression equation is f( t) 0.0t.8. This equation will be used for f in parts b e.) The graph of the model fits the data well. b. The slope of the model is 0.0. This means that the percentage of the military who are women is increasing by 0.0% each year. c. f(100) 0.0(100) This means that 15. percent of the military will be women in the year 000 (t 100). d. Let f (t) 100 and solve for t t t 0.0 t 1.05 This means that 100 percent of the military will be women in the year e. Model breakdown occurs for certain when the percent is below 0 or above 100. Find t when f (t) t t t t 6.05 Therefore, model breakdown occurs in years before , since it is not possible for a percent to be negative in this context. We know that, according to the model, 100 percent of the military will be women in 1. Therefore, model breakdown is occurring in years after 1. Therefore, model breakdown occurs when t < 6.05 and t > Use the first three steps of the modeling process to find the equation. Use those points 5 (1, 11.) and (5, 6.) to find the slope of the line m So f (t) 9.1t + b. To solve for b, substitute 1 for t and 11. for f (t) since the line contains (1, 11.) (1) + b b.0 b The equation for f is f (t) 9.1t +.0. Predicting for 010 using this equation, let t 10 for f (t). f (10) 9.1(10) This means that in 010, total ad spending for the NCAA basketball tournament can be predicted to be $66. million. 1. a. Yes, the rate of increase has remained constant, showing a linear relationship. The slope in this case is 6.8, or an increase in 6.8 percentage points per year. b. The p-intercept is the p value when t 0, or during 001. In 001, p 1, or 1% of community banks offered Internet banking in 001. c. Based on the first two parts, the equation of the model is f(t) 6.8t + 1. d. Since , t corresponds to 008. f () 6.8() The model predicts that in 008, 88.6 percent of community banks will offer Internet banking. e. To find when all community banks will offer Internet banking, substitute 100 for p, and solve for t t t 9 t The model predicts that in 010 all

21 community banks will offer Internet banking.. Answers may vary. 5