4.3 Modeling with Arithmetic Sequences

Transcription

1 Name Class Date 4.3 Modelig with Arithmetic Sequeces Essetial Questio: How ca you solve real-world problems usig arithmetic sequeces? Resource Locker Explore Iterpretig Models of Arithmetic Sequeces You ca model real-world situatios ad solve problems usig models of arithmetic sequeces. For example, suppose watermelos cost $6. each at the local market. The total cost, i dollars, of watermelos ca be foud usig c() = 6.5. A Complete the table of values for 1, 2, 3, ad 4 watermelos. Watermelos Total cost ($) c() B What is the commo differece? C D E What does represet i this cotext? What are the depedet ad idepedet variables i this cotext? Fid c(7). What does this value represet? Reflect 1. Discussio What domai values make sese for c() = 6.5 i this situatio? Module Lesso 3

2 Explai 1 Modelig Arithmetic Sequeces From a Table Give a table of data values from a real-world situatio ivolvig a arithmetic sequece, you ca costruct a fuctio model ad use it to solve problems. Example 1 A Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the table. The iterpret the meaig of a specific term of the sequece i the give cotext. Suppose the table shows the cost, i dollars, of postage per ouce of a letter. Number of ouces Cost ($) of postage B Determie the value of ƒ (9), ad tell what it represets i this situatio. Fid the commo differece, d. d = =.2 Substitute.35 for ƒ (1) ad.2 for d. ƒ () = ƒ (1) + d( - 1) ƒ () = ( - 1) ƒ (9) = (8) = 1.95 So, the cost of postage for a 9-ouce letter is $1.95. The table shows the cumulative total iterest paid, i dollars, o a loa after each moth. Number of moths Cumulative total ($) Determie the value of ƒ (2) ad tell what it represets i this situatio. Fid the commo differece, d. d = - 16 = Substitute for ƒ (1) ad for d. ƒ () = ƒ (1) + d( - 1) ƒ () = + ( - 1) Fid ƒ (2) ad iterpret the value i cotext. ƒ () = ƒ (1) + d( - 1) ƒ ( ) = + ( ) = So, the cumulative total paid after moths is. Your Tur Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the table. The iterpret the meaig of a specific term of the sequece i the give cotext. 2. The table shows ƒ (), the distace, i miles, from the store after Mila has traveled for hours. Time (h) Distace (mi) Determie the value of ƒ (1) ad tell what it represets i this situatio. Module Lesso 3

3 3. The table below shows the total cost, i dollars, of purchasig battery packs. Number of battery packs Total cost ($) Determie the value of ƒ (18) ad tell what it represets i this situatio. Explai 2 Modelig Arithmetic Sequeces From a Graph Give a graph of a real-world situatio ivolvig a arithmetic sequece, you ca costruct a fuctio model ad use it to solve problems. Example 2 Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the graph, ad use it to solve the problem. A D Adre collects feather pes. The graph shows the umber of feather pes D Adre has collected over time, i weeks. Accordig to this patter, how may feather pes will D Adre have collected i 12 weeks? Represet the sequece i a table f ( ) Fid the commo differece. Number of feather pes y (4, 75) (3, 56) (2, 37) (1, 18) x Time (weeks) d = 37-8 = 19 Use the geeral explicit rule for a arithmetic sequece to write the rule i fuctio otatio. Substitute 18 for ƒ (1) ad 19 for d. ƒ () = ƒ (1) + d( - 1) ƒ () = ( - 1) To determie the umber of feather pes D Adre will have collected after 12 weeks, fid ƒ(12). ƒ () = ( - 1) ƒ (12) = (11) ƒ (12) = ƒ (12) = 227 So, if this patter cotiues, D Adre will have collected 227 feather pes i 12 weeks. Module Lesso 3

4 B Eric collects stamps. The graph shows the umber of stamps that Eric has collected over time, i moths. Accordig to this patter, how may stamps will Eric have collected i 1 moths? Represet the sequece i a table Number of stamps y (4, 59) (3, 46) (2, 33) (1, 2) x Time (moths) Fid the commo differece. d = - 2 = Use the geeral explicit rule for a arithmetic sequece to write the rule i fuctio otatio. Substitute for ƒ(1) ad for d. ƒ () = ƒ (1) + d ( - 1) ƒ () = + ( - 1) To determie the umber of stamps Eric will have collected i 1 moths, fid ƒ ( ). ƒ () = ƒ (1) + d ( - 1) ƒ ( ) = + ( ) = So, if this patter cotiues, Eric will have collected i moths. Reflect 4. How do you kow which variable is the idepedet variable ad which variable is the depedet variable i a real-world situatio ivolvig a arithmetic sequece? Your Tur Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the graph, ad use it to solve the problem. 5. The graph shows the height, i iches, of a stack of boxes o a table as the umber of boxes i the stack icreases. Fid the height of the stack with 7 boxes. Height (i.) y (1, 34) (3, 68) (2, 51) (4, 85) x Number of boxes Module Lesso 3

5 6. Quyh begis to save the same amout each moth to save for a future shoppig trip. The graph shows total amout she has saved after each moth,. What will be the total amout Quyh has saved after 12 moths? f f ( ) Amout saved ($) y (4, 4) (3, 3) (2, 3) (1, 2) x Number of moths Explai 3 Modelig Arithmetic Sequeces From a Descriptio Give a descriptio of a real-world situatio ivolvig a arithmetic sequece, you ca costruct a fuctio model ad use it to solve problems. Example 3 Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted, ad use it to solve the problem. Justify ad evaluate your aswer. The odometer o a car reads 34,24 o Day 1. Every day the car is drive 57 miles. If this patter cotiues, what will the odometer read o Day 15? Aalyze Iformatio The odometer o the car reads miles o Day 1. Every day the car is drive miles. ƒ (1) = d = ad Formulate a Pla Write a explicit rule i fuctio otatio for the arithmetic sequece, ad use it to fid, the odometer readig o Day 15. Solve ƒ () = ƒ (1) + d ( - 1) ƒ () = + ( - 1) ƒ ( ) = + ( ) ƒ ( ) = O the Day 15, the odometer will show miles. Module Lesso 3

6 Justify ad Evaluate Usig a arithmetic sequece model reasoable because the umber of miles o the odometer icreases by the same amout each day. By roudig ad estimatio: 34,2 + 6 (14) = + = miles So miles is a reasoable aswer. Your Tur Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted, ad use it to solve the problem. Justify ad evaluate your aswer. 7. Ruby siged up for a frequet-flier program. She receives 34 frequet-flier miles for the first roud-trip she takes ad 12 frequet-flier miles for all additioal roud-trips. How may frequet-flier miles will Ruby have after 5 roud-trips? 8. A gym charges each member $1 for the first moth, which icludes a membership fee, ad $ per moth for each moth after that. How much moey will a perso sped o their gym membership for 6 moths? Elaborate 9. What domai values usually make sese for a arithmetic sequece model that represets a real-world situatio? 1. Whe give a graph of a arithmetic sequece that represets a real-world situatio, how ca you determie the first term ad the commo differece i order to write a model for the sequece? 11. What are some ways to justify your aswer whe creatig a arithmetic sequece model for a real-world situatio ad usig it to solve a problem? 12. Essetial Questio Check-I How ca you costruct a model for a real-world situatio that ivolves a arithmetic sequece? Module 4 18 Lesso 3

7 Evaluate: Homework ad Practice 1. A T-shirt at a departmet store costs $7.. The total cost, i dollars, of a T-shirts is give by the fuctio C(a) = 7.5a. a. Complete the table of values for 4 T-shirts. T-shirts Cost ($) b. Determie the commo differece. c. What does the variable a represet? What are the reasoable domai values for a? 2. A car dealership sells 5 cars per day. The total umber of cars C sold over time i days is give by the fuctio C(t) = 5t. a. Complete the table of values for the first 4 days of sales. Time (days) Number of Cars b. Determie the commo differece. c. What do the variables represet? What are the reasoable domai ad rage values for this situatio? 3. A telemarketer makes 82 calls per day. The total umber of calls made over time, i days, is give by the fuctio C(t) = 82t. a. Complete the table of values for 4 days of calls. Time (days) Number of Calls b. Determie the commo differece. c. What do the variables represet? What are the reasoable domai ad rage values for this situatio? Module Lesso 3

8 Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the table. The determie the value of the give term, ad explai what it meas. 4. Darell starts savig the same amout from each week s paycheck. The table shows the total balace ƒ() of his savigs accout over time i weeks. Time (weeks) Savigs Accout Balace($) $38 $51 $64 Determie the value of ƒ (9), ad explai what it represets i this situatio. 5. Jua is travelig to visit uiversities. He otices mile markers alog the road. He records the mile markers every 1 miutes. His father is drivig at a costat speed. Complete the table. a. Time Iterval Mile Marker b. Fid ƒ(1), ad tell what it represets i this situatio. Costruct a explicit rule i fuctio otatio for the arithmetic sequece represeted i the graph. The determie the value of the give term, ad explai what it meas. 6. The graph shows total cost of a whitewater raftig trip ad the correspodig umber of passegers o the trip. Fid ƒ(8), ad explai what it represets. Number of Passegers Total Cost ($) f() 7. Ed collects autographs. The graph shows the total umber of autographs that Ed has collected over time, i weeks. Fid ƒ(12), ad explai what it represets. Total cost ($) Number of autographs f() (4, 1) (3, 125) (2, 1) (1, 75) Number of passegers f() (2, 35) (1, 2) (4, 65) (3, ) Time (weeks) (5, 8) Module Lesso 3

9 8. Fiace Bob purchased a bus pass card with 32 poits. Each week costs 2 poits for ulimited bus rides. The graph shows the poits remaiig o the card over time i weeks. Determie the value of ƒ(1), ad explai what it represets. Poits remaiig f() (2, 28) (1, 3) (3, 26) (4, 24) (5, 22) Time (weeks) 9. Biology The local wolf populatio is decliig. The graph shows the local wolf populatio over time, i weeks. Fid ƒ(9), ad explai what it represets. Number of wolves f() (1, 1) (2, 94) (3, 88) (4, 82) (5, 76) Time (weeks) Costruct a explicit rule i fuctio otatio for the arithmetic sequece. The determie the value of the give term, ad explai what it meas. 1. Ecoomics To package ad ship a item, it costs $5.75 for the first poud ad $.75 for each additioal poud. Fid the 12th term, ad explai what it represets. 11. A ew bag of cat food weighs 18 pouds. At the ed of each day,.5 poud of food is removed to feed the cats. Fid the 3th term, ad explai what it represets. Module Lesso 3

10 12. Carrie borrows $96 iterest-free to pay for a car repair. She will repay $12 mothly util the loa is paid off. How may moths will it take Carrie to pay off the loa? Explai. 13. The rates for a go-kart course are show. a. What is the total cost for 15 laps? Number of Laps Total cost ($) f() b. Suppose that after payig for 9 laps, the 1th lap is free. Will the sequece still be arithmetic? Explai. 14. Multi-Part Seats i a cocert hall are arraged i the patter show. a. The umbers of seats i the rows form a arithmetic sequece. Write a rule for the arithmetic sequece. Row 1 Row 2 Row 3 b. How may seats are i Row 15? Row 4 c. Each ticket costs $4. If every seat i the first 1 rows is filled, what is the total reveue from those seats? d. A extra chair is added to each row. Write the ew rule for the arithmetic sequece ad fid the ew total reveue from the first 1 rows. Module Lesso 3

11 H.O.T. Focus o Higher Order Thikig 15. Explai the Error The table shows the umber of people who atted a amusemet park over time, i days. Time (days) Number of people f() Sam writes a explicit rule for this arithmetic sequece: ƒ() = ( - 1) He the claims that accordig to this patter, 325 people will atted the amusemet park o Day 5. Explai the error that Sam made. 16. Commuicate Mathematical Ideas Explai why it may be harder to fid the th value of a arithmetic sequece from a graph if the poits are ot labeled. 17. Make a predictio Veroa is traiig for a maratho. The first part of her traiig schedule is give i the table. Sessio Distace (mi) a. Is this traiig schedule a arithmetic sequece? Explai. If it is, write a explicit rule for the sequece. b. If Veroa cotiues this patter, durig which traiig sessio will she ru 26 miles? 18. If Veroa s traiig schedule starts o a Moday ad she rus every third day, o which day will she ru 26 miles? Module Lesso 3

12 19. Multiple Represetatios Determie whether the followig graph, table, ad verbal descriptio all represet the same arithmetic sequece. Time (moths) Amout of moey ($) A perso deposits $2 dollars ito a bak accout. Each moth, he adds $25 dollars to the accout, ad o other trasactios occur i the accout. Amout of moey ($) f() (4, 4) (3, 3) (2, 3) (1, 2) Time (moths) Lesso Performace Task The graph shows the populatio of Ivor s at coloy over the first four weeks. Assume the at populatio will cotiue to grow at the same rate. a. Write a explicit rule i fuctio otatio. Ivor s At Farm b. If Ivor s ats have a mass of 1.5 grams each, what will be the total mass of all of his ats i 13 weeks? Number of Ats Weeks c. Whe the coloy reaches 1385 ats, Ivor s at farm will ot be big eough for all of them. I how may weeks will the at populatio be too large? Module Lesso 3