2.3 Formulas. Evaluating Formulas. EXAMPLE 1 Event promotion. Event promoters use the formula. Evaluating Formulas a Solving for a Variable

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1 2.3 Formula Evaluating Formula a Solving for a Variable Many application of matematic involve relationip among two or more quantitie. An equation tat repreent uc a relationip will ue two or more letter and i known a a formula. Mot of te letter in ti book are variable, but ome are contant. For example, c in E = mc 2 repreent te peed of ligt. Evaluating Formula EXAMPLE 1 Event promotion. Event promoter ue te formula p = 1.2x to determine a ticket price p for an event wit x dollar of expene and anticipated ticket ale. Grand Event expect expene for an upcoming concert to be $80,000 and anticipate elling 4000 ticket. Wat ould te ticket price be? Source: Te Indianapoli Star, 2/27/03

2 94 CHAPTER 2 EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING SOLUTION We ubtitute 80,000 for x and 4000 for in te formula and calculate p: p = 1.2x = , = 24. Te ticket price ould be $24. TRY EXERCISE 1 Solving for a Variable In te Norteat, te formula B = 30a i ued to determine te minimum furnace output B, in Briti termal unit (Btu ), for a well-inulated ome wit a quare feet of flooring. Suppoe tat a contractor a an extra furnace and want to determine te ize of te larget (well-inulated) oue in wic it can be ued. Te contractor can ubtitute te amount of te furnace output in Btu ay, 63,000 for B, and ten olve for a: 63,000 = 30a Replacing B wit 63, = a. Dividing bot ide by 30 Te ome ould ave no more tan 2100 ft 2 of flooring. Were tee calculation to be performed for a variety of furnace, te contractor would find it eaier to firt olve B = 30a for a, and ten ubtitute value for B. Solving for a variable can be done in muc te ame way tat we olved equation in Section 2.1 and 2.2. TECHNOLOGY CONNECTION Suppoe tat after calculating 63,000, 30, we wi to find 72,000, 30. Preing Fvgive te following / / / /30 EXAMPLE Moving te curor left, we can cange 63,000 to 72,000 and pre [ Solve for a: B = 30a. SOLUTION We ave B = 30a B 30 = a. We want ti letter alone. Dividing bot ide by 30 Te equation a = B>30 give a quick, eay way to determine te floor area of te larget (well-inulated) oue tat a furnace upplying B Btu could eat. TRY EXERCISE 9 To ee ow olving a formula i jut like olving an equation, compare te following. In (A), we olve a uual; in (B), we ow tep but do not implify; and in (C), we cannot implify becaue a, b, and c are unknown. A. 5 x + 2 = 12 B. 5 x = x = 10 x = 10 5 = 2 5 x + 2 = 12 C. 5 x = 12-2 x = ax + b = c ax = c - b x = c - b a 1. Verify te work above and ten ue Fv to find 72,000,

3 2.3 FORMULAS 95 EXAMPLE 3 Circumference of a circle. Te formula C = 2pr give te circumference C of a circle wit radiu r. Solve for r. SOLUTION Te circumference i te ditance around a circle. r Given a radiu r, we can ue ti equation to find a circle circumference C. Given a circle circumference C, we can ue ti equation to find te radiu r. C = 2pr C 2p = 2pr 2p C 2p = r We want ti variable alone. Dividing bot ide by 2p TRY EXERCISE 13 EXAMPLE 4 Solve for y: 3x - 4y = 10. SOLUTION Tere i one term tat contain y, o we begin by iolating tat term on one ide of te equation. 3 x - 4y = 10 We want ti variable alone. -4y = 10-3x Subtracting 3x from bot ide y2 = x2 Multiplying bot ide by - 4 y = x Multiplying uing te ditributive law y = x Simplifying te fraction TRY EXERCISE 33 STUDY SKILLS Pace Yourelf EXAMPLE 5 Mot intructor agree tat it i better for a tudent to tudy for one our four day in a week, tan to tudy once a week for four our. Of coure, te total weekly tudy time will vary from tudent to tudent. It i common to expect an average of two our of omework for eac our of cla time. Nutrition. Te number of calorie K needed eac day by a moderately active woman wo weig w pound, i ince tall, and i a year old, can be etimated uing te formula Solve for w. SOLUTION K = w + - a2. * We revere te order in wic te operation occur on te rigt ide: We want w alone. K = w + - a2 K = 61w + - a2 Subtracting 917 from bot ide K = w + - a Dividing bot ide by 6 6 K Adding a and ubtracting on + a - = w. 6 bot ide Ti formula can be ued to etimate a woman weigt, if we know er age, eigt, and caloric need. TRY EXERCISE 43 *Baed on information from M. Parker (ed.), Se Doe Mat! (Waington, D.C.: Matematical Aociation of America, 1995), p. 96.

4 96 CHAPTER 2 EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Te above tep are imilar to toe ued in Section 2.2 to olve equation. We ue te addition and multiplication principle jut a before. An important difference tat we will ee in te next example i tat we will ometime need to factor. To Solve a Formula for a Given Variable 1. If te variable for wic you are olving appear in a fraction, ue te multiplication principle to clear fraction. 2. Iolate te term(), wit te variable for wic you are olving on one ide of te equation. 3. If two or more term contain te variable for wic you are olving, factor te variable out. 4. Multiply or divide to olve for te variable in quetion. We can alo olve for a letter tat repreent a contant. EXAMPLE 6 r Surface area of a rigt circular cylinder. Te formula A = 2pr + 2pr 2 give te urface area A of a rigt circular cylinder of eigt and radiu r. Solve for p. SOLUTION We ave A = 2pr + 2pr 2 A = p12r + 2r 2 2 A 2r + 2r 2 = p. We can alo write ti a p = A 2r + 2r 2. We want ti letter alone. Factoring Dividing bot ide by 2r + 2r 2, or multiplying bot ide by 1>12r + 2r 2 2 TRY EXERCISE 47 CAUTION! Had we performed te following tep in Example 6, we would not ave olved for p: A = 2pr + 2pr 2 We want p alone. A - 2pr 2 = 2pr Subtracting 2pr 2 from bot ide A - 2pr 2 2r = p. Two occurrence of p Dividing bot ide by 2r Te matematic of eac tep i correct, but becaue p occur on bot ide of te formula, we ave not olved te formula for p. Remember tat te letter being olved for ould be alone on one ide of te equation, wit no occurrence of tat letter on te oter ide!

5 2.3 FORMULAS EXERCISE SET For Extra Help 1. Outdoor concert. Te formula d = 344t can be ued to determine ow far d, in meter, ound travel troug room-temperature air in t econd. At a large concert, fan near te back of te crowd experienced a 0.9-ec time lag between te time eac word wa pronounced on tage (a own on large video monitor) and te time te ound reaced teir ear. How far were tee fan from te tage? m were I i te core inflation rate over te previou 12 mont and U i te eaonally adjuted unemployment rate. If core inflation i and unemployment i 0.044, wat ould te federal fund rate be? Source: Greg Mankiw, Harvard Univerity, 6. Calorie denity. Te calorie denity D, in calorie per ounce, of a food tat contain c calorie and weig w ounce i given by D = c w.* 2. Furnace output. Contractor in te Norteat ue te formula B = 30a to determine te minimum furnace output B, in Briti termal unit (Btu ), for a well-inulated oue wit a quare feet of flooring. Determine te minimum furnace output for an 1800-ft 2 oue tat i well inulated. 54,000 Btu Source: U.S. Department of Energy 3. College enrollment. At many college, te number of full-time-equivalent tudent f i given by f = n 15, were n i te total number of credit for wic tudent ave enrolled in a given emeter. Determine te number of full-time-equivalent tudent on a campu in wic tudent regitered for a total of 21,345 credit tudent 4. Ditance from a torm. Te formula M = 1 5 t can be ued to determine ow far M, in mile, you are from ligtning wen it tunder take t econd to reac your ear. If it take 10 ec for te ound of tunder to reac you after you ave een te ligtning, ow far away i te torm? 2 mi 5. Federal fund rate. Te Federal Reerve Board et a target f for te federal fund rate, tat i, te interet rate tat bank carge eac oter for overnigt borrowing of Federal fund. Ti target rate can be etimated by f = I - U2, Eigt ounce of fat-free milk contain 84 calorie. Find te calorie denity of fat-free milk cal>oz 7. Aborption of ibuprofen. Wen 400 mg of te painkiller ibuprofen i wallowed, te number of milligram n in te bloodtream t our later (for 0 t 6) i etimated by n = 0.5t t t t. How many milligram of ibuprofen remain in te blood 1 r after 400 mg a been wallowed? 255 mg 8. Size of a league cedule. Wen all n team in a league play every oter team twice, a total of N game are played, were N = n 2 - n. If a occer league a 7 team and all team play eac oter twice, ow many game are played? 42 game In Exercie 9 48, olve eac formula for te indicated letter. 9. A = b, for b (Area of parallelogram wit bae b and eigt ) b = A b 10. A = b, for = A b 11. d = rt, for r (A ditance formula, were d i ditance, r i peed, and t i time) r = d t *Source: Nutrition Action Healtletter, Marc 2000, p. 9. Center for Science in te Public Interet, Suite 300; 1875 Connecticut Ave NW, Waington, D.C

6 98 CHAPTER 2 EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING 12. d = rt, for t t = d r 13. I = Prt, for P (Simple-interet formula, were I i interet, P i principal, r i interet rate, and t i time) 14. I = Prt, for t t = I Pr P = I rt 15. H = 65 - m, for m (To determine te number of eating degree day H for a day wit m degree Fareneit a te average temperature) m = 65 - H 16. d = - 64, for (To determine ow many ince d above average an -inc-tall woman i) = d P = 2l + 2w, for l (Perimeter of a rectangle of lengt l and widt w) l = P - 2w, or l = P 2 w 2 - w 18. P = 2l + 2w, for w w = P - 2l, or w = P l 19. A = pr 2, for p (Area of a circle wit radiu r) p = A r for r 2 A A = pr 2, r 2 = p 21. A = 1 2 b, for (Area of a triangle wit bae b and eigt ) = 2A b 22. A = 1 2 b, for b b = 2A 23. E = mc 2, for c 2 (A relativity formula from pyic) 24. E = mc 2, for m 25. Q = c + d for d 26. Q = p - q, for p 2, 2 d = 2Q - c p = 2Q + q 27. A = a + b + c, for b 28. A = a + b + c, for c 3 3 b = 3A - a - c c = 3A - a - b 9 Anwer to Exercie can be found on p. IA-1. b m = E c 2 l r c 2 = E m 29. w = r, for r f (To compute te wavelengt w of a muical note wit frequency f and peed of ound r) r = wf 30. M = A for A, (To compute te Mac number M for peed A and peed of ound ) A = M 31. F = 9 for C = 5 1F C + 32, C 9 (To convert te Celiu temperature C to te Fareneit temperature F ) 32. M = 5, for 9 n + 18 n 33. 2x - y = 1, for y 34. 3x - y = 7, for y y = 3x x + 5y = 10, for 2 y = x + 2y = 12, for 3 y = x - 3y = 6 4, for y = 3 x x - 4y = 8, for y = 5 4 x x + 8y = 4, for y 40. x + 10y = 2, for y 41. 3x - 5y = 8, for y 42. 7x - 6y = 7, for y y = 2x - 1 y = x y = x y = 3 5 x y = 7 6 x z = x + y2, for x A = p + 2, for t = w - l2, for l 9 4 n = 9 1M m = 19-51x - n2, for n 9 A 47. A = at + bt, for t t = a + b S 48. S = rx + x, for x x = r Area of a trapezoid. Te formula A = 1 2 a b can be ued to find te area A of a trapezoid wit bae a and b and eigt. Solve for. (Hint: Firt clear fraction.) 2A = a + b a b

7 2.3 FORMULAS 99 Aa! 50. Compounding interet. Te formula A = P + Prt i ued to find te amount A in an account wen imple interet i added to an invetment of P dollar (ee Exercie 13). Solve for P. A P = 1 + rt 51. Ce rating. Te formula 4001W - L2 R = r + N i ued to etabli a ce player rating R after tat player a played N game, won W of tem, and lot L of tem. Here r i te average rating of te opponent. Solve for L. Source: Te U.S. Ce Federation 52. Angle meaure. Te angle meaure S of a ector of a circle i given by S = 360A pr 2, were r i te radiu, A i te area of te ector, and S i in degree. Solve for r 2. r 2 = 360A Sp 53. Naomi a a formula tat allow er to convert Celiu temperature to Fareneit temperature. Se need a formula for converting Fareneit temperature to Celiu temperature. Wat advice can you give er? 54. Under wat circumtance would it be ueful to olve d = rt for r? (See Exercie 11.) Skill Review Review implifying expreion (Section 1.6, 1.7, and 1.8). Perform te indicated operation (-4) - 17 [1.6] , 1 [1.7] (-11.75)(0) [1.7] (-2) 5 [1.8] -32 Simplify. [1.8] , 1-42 # S r Syntei 61. Te equation P = 2l + 2w and w = P 2 - l are equivalent formula involving te perimeter P, lengt l, and widt w of a rectangle. Devie a problem for wic te econd of te two formula would be more ueful. 62. Wile olving 2A = a + b for, Lea write 2A - a =. Wat i er mitake? b 63. Te Harri Benedict formula give te number of calorie K needed eac day by a moderately active man wo weig w kilogram, i centimeter tall, and i a year old a K = w a If Jano i moderately active, weig 80 kg, i 190 cm tall, and need to conume 2852 calorie a day, ow old i e? 40 yr 64. Altitude and temperature. Air temperature drop about 1 Celiu (C) for eac 100-m rie above ground level, up to 12 km. If te ground level temperature i t C, find a formula for te temperature T at an elevation of meter. Source: A Sourcebook of Scool Matematic, Matematical Aociation of America, 1980 T = t C 65. Surface area of a cube. Te urface area A of a cube wit ide i given by A = 6 2. If a cube urface area i 54 in 2, find te volume of te cube. 27 in Weigt of a fi. An ancient fierman formula for etimating te weigt of a fi i w = lg2 800, were w i te weigt, in pound, l i te lengt, in ince, and g i te girt (ditance around te midection), in ince. Etimate te girt of a 700-lb yellow tuna tat i 8 ft long. About 76.4 in. N1R - r2 51. L = W -, or L = W - NR + Nr 400

8 100 CHAPTER 2 EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING 67. Doage ize. Clark rule for determining te ize of a particular cild medicine doage c i c = w a # d, were w i te cild weigt, in pound, and d i te uual adult doage for an adult weiging a pound. Solve for a. Source: Olen, June Looby, et al., Medical Doage Calculation. Redwood City, CA: Addion-Weley, 1995 a = w # d c Solve eac formula for te given letter. y 68. for y y = z2 z, z t = 1, t d 69. ac = bc + d, for c c = a - b r 70. qt = r1 + t2, for t t = q - r c 71. 3a = c - a1b + d2, for a a = 3 + b + d 72. Furnace output. Te formula B = 50a i ued in New England to etimate te minimum furnace output B, in Btu, for an old, poorly inulated oue wit a quare feet of flooring. Find an equation for determining te number of Btu aved by inulating an old oue. (Hint: See Exercie 2.) S = 20a, were S i te number of Btu aved 73. Revie te formula in Exercie 63 o tat a man weigt in pound lb = 1 kg2 and i eigt in ince in. = 1 cm2 are ued. K = 9.632w a Revie te formula in Example 5 o tat a woman weigt in kilogram lb = 1 kg2 and er eigt in centimeter in. = 1 cm2 are ued. K = w a