1.3 Using Formulas to Solve Problems
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1 Section 1.3 Uing Fomul to Solve Polem Uing Fomul to Solve Polem OBJECTIVES 1 Solve fo Vile in Fomul 2 Ue Fomul to Solve Polem Peping fo Fomul Befoe getting tted, tke ti edine quiz. If you get polem ong, go ck to te ection cited nd evie te mteil. In Polem P1 nd P2, () ound, nd () tuncte ec deciml to te indicted nume of plce. [Section R.2, pp ] P ; 3 deciml plce P ; 2 deciml plce A knon eltion tt exit eteen to o moe vile cn e ued to olve cetin type of polem. A fomul i n eqution tt decie o to o moe vile e elted. Wok Smt Tink of fomul note kind of mtemticl model. EXAMPLE 1 Aneing Quetion it Fomul Te infomtion in Tle 5 give deciption, in od, of knon eltion nd te coeponding fomul. Vel Deciption Tle 5 Geomety: Wt i te e of ectngle? Te e A of ectngle i te poduct of it lengt l nd idt. Pyic: Ho do e meue te enegy of n oject in motion? Kinetic enegy K i one-lf te poduct of te m m nd te que of te velocity v. Economic: Wt popotion of te popultion i in te lo foce? Te pticiption te R i te um of te nume of employed E nd te nume of unemployed U divided y te dult popultion P. Finnce: Ho muc money ill I ve next ye? Te futue vlue A i te poduct of te peent vlue P nd 1 plu te nnul inteet te. Fomul A = l K = 1 2 mv2 R = E + U P A = P A i n eqution tt decie o to o moe vile e elted. In Polem 2 5, tnlte te vel deciption into mtemticl fomul. 2. Te e A of cicle i te poduct of te nume p nd te que of it diu. 3. Te volume V of igt cicul cylinde i te poduct of te nume p, te que of it diu, nd it eigt. 4. Te dily cot C of mnufctuing compute i $175 time te nume of compute mnufctued x plu $ Te ditnce tt n oject fee-fll i one-lf te poduct of cceletion due to gvity g nd te que of time t. Peping fo...ane P1. () () P2. () () Solve fo Vile in Fomul Te expeion olve fo te vile men to get te vile y itelf on one ide of te eqution it ll ote vile nd contnt, if ny, on te ote ide y foming equivlent eqution. Fo exmple, in te fomul fo te e of ectngle,
2 74 CHAPTER 1 Line Eqution nd Inequlitie A = l, te fomul i olved fo A ecue A i y itelf on one ide of te eqution ile ll ote vile e on te ote ide. Wen olving cetin polem, it ecome impotnt fo u to e le to olve fomul fo cetin vile. Te tep tt e follo en olving fomul fo cetin vile e identicl to toe tt e folloed en olving n eqution. EXAMPLE 2 Solving fo Vile in Fomul Te volume V of cone i given y te fomul V = 1 ee i te diu nd i te eigt of te cone. See Figue 4. 3 p2, () Solve te fomul fo. () Ue te eult fom pt () to find te eigt of cone if it volume i 50p cuic feet nd it diu i 5 feet. Figue 4 Wok Smt Solving fo vile i jut like olving n eqution it one unknon. Wen olving fo vile, tet ll te ote vile contnt. Wok Smt Wen oking it fomul, keep tck of te unit. Veify tt te unit in you ne e eonle. () Becue e nt to olve te fomul fo, e nt to get y itelf nd ll ote vile nd contnt on te ote ide of te eqution. Multiply ot ide y 3: Divide out common fcto: Divide ot ide y p 2 : Divide out common fcto: If, ten : () Sutituting V = 50p nd ft into = 3V ft3 = 5 e otin p 2, = 3150p ft3 2 p15 ft p ft3 = 25p ft 2 = 6 feet V = 1 3 p2 3 # V = 1 3 p2 # 3 3V = p 2 3V p 2 = p2 p 2 3V p 2 = = 3V p 2 Exmple 2 peent fomul fom geomety. Fomul fom geomety e ueful in olving mny type of polem. We lit ome of tee fomul in Tle 6. Tle 6 Figue Fomul Figue Fomul Sque Ae: A = 2 Peimete: P = 4 Rectngle Ae: A = l Peimete: P = 2l + 2 l
3 Section 1.3 Uing Fomul to Solve Polem 75 Figue Fomul Figue Fomul Rectngul Solid Volume: V = l Tingle Ae: A = 1 2 Sufce Ae: S = 2l + 2l + 2 Peimete: P = + + c c l Tpezoid Ae: A = 1 1B Peimete: P = + + c + B Spee Volume: V = 4 3 p 3 Sufce Ae: S = 4p 2 c B Pllelogm Ae: A = Peimete: P = Rigt Cicul Cylinde Volume: V = p 2 Sufce Ae: S = 2p 2 + 2p Cicle Ae: A = p 2 Cicumfeence: C = 2p = pd Cone Volume: V = 1 3 p2 d Cue Volume: V = 3 Sufce Ae: S = Te fomul fo te e of cicle i. 7. Wt i te fomul fo te peimete of ectngle? 8. Te e A of tingle i given y te fomul A = 1 ee i te e of te tingle nd i te eigt. 2, () Solve te fomul fo. () Find te eigt of te tingle oe e i 10 que ince nd oe e i 4 ince. 9. Te peimete P of pllelogm i given y te fomul P = 2 + 2, ee i te lengt of one ide of te pllelogm nd i te lengt of te djcent ide. () Solve te fomul fo. () Find te lengt of one ide of pllelogm oe peimete i 60 cm nd oe djcent lengt i 20 cm.
4 76 CHAPTER 1 Line Eqution nd Inequlitie EXAMPLE 3 Solving fo Vile in Fomul Te fomul Y = C + Y + I + G + N i model ued in economic to decie te totl income of n economy. In te model, Y i income, C i conumption, I i invetment in cpitl, G i govenment pending, N i net expot, nd i contnt. Solve te fomul fo Y. We nt to get ll tem it Y on te me ide of te equl ign. Y = C + Y + I + G + N Sutct Y fom ot ide: Y - Y = C + Y + I + G + N - Y Comine like tem: Y - Y = C + I + G + N Ue te Ditiutive Popety in evee to iolte Y: Y11-2 = C + I + G + N Y11-2 Divide ot ide y 1 : = C + I + G + N 1 - Simplify: Y = C + I G + N 1 - In Polem 10 13, olve fo te indicted vile. 10. I = Pt fo P 11. Ax + By = C fo y 12. 2x - 4x = 3-3 fo 13. S = n + 1n - 12d fo n 2 Ue Fomul to Solve Polem Fomul e often needed in ode to olve cetin type of od polem. We follo te me tep to olving tee polem ee peented in Section 1.2 on pge 61. EXAMPLE 4 Te Peimete of Windo Te peimete of ectngul pictue indo i 466 ince. Te lengt of te indo i 55 ince moe tn te idt. See Figue 5. Find te dimenion of te indo. Step 1: Identify Ti i geomety polem tt equie te fomul fo te peimete of ectngle. We nt to detemine te dimenion of te indo, ic i in te pe of ectngle.tt i, e nt to detemine te lengt nd idt of te indo. Figue 5 Step 2: Nme Let epeent te idt. Since te lengt i 55 ince moe tn te idt, e kno tt l = Step 3: Tnlte Te peimete of ectngle i P = 2l + 2. We utitute te knon vlue into te fomul fo te peimete of ectngle. P = 466; l = + 55: P = 2l = Te Model Step 4: Solve 466 = Ditiute te 2: 466 = Comine like tem: 466 = Sutct 110 fom ot ide: 356 = 4 Divide ot ide y 4: 89 =
5 Section 1.3 Uing Fomul to Solve Polem 77 Step 5: Ceck It ppe tt te idt of te indo i 89 ince, o te lengt i = 144 ince. Te peimete of te indo i = 466 ince. It ceck! Step 6: Ane te Quetion Te idt of te indo i 89 ince nd te lengt i 144 ince. Tee i n inteeting ide note to te eult of Exmple 4. If e compute te tio 144 of te lengt of te indo to te idt, e otin Rectngle oe 89 L dimenion fom ti tio e clled golden ectngle. Golden ectngle e id to ve dimenion tt e pleing to te eye. Te golden ectngle fit contucted y te Geek piloope Pytgo in te ixt centuy B.C. Tee ectngle e ued in citectue (Te Ptenon) nd in t (te Mon Li). See Figue 6. Figue 6 Dgli Oti (A)/Pictue Dek, Inc./ Kol Collection 14. Te peimete of ectngul pool i 180 feet. If te lengt of te pool i to e 10 feet moe tn te idt, find te dimenion of te pool. 15. Te opening of ectngul ookce peimete of 224 ince. If te eigt of te ookce i 32 ince moe tn te idt, detemine te dimenion of te opening of te ookce. Figue 7 EXAMPLE 5 Contucting Soup Cn A cn of Cmpell oup ufce e of 46.5 que ince. See Figue 7. Te ufce e S of igt cicul cylinde i S = 2p 2 + 2p, ee i te diu of te cn nd i te eigt of te cn. Find te eigt of cn of Cmpell oup if it diu i ince. Round you ne to to deciml plce Step 1: Identify Ti i geomety polem tt equie te fomul fo te ufce e of cylinde. We i to find te eigt of te cn of oup. Step 2: Nme Let epeent te eigt of te cn of oup. Step 3: Tnlte We kno tt te ufce e S of te cn of oup i 46.5 que ince. Te diu of te cn of oup i ince. We utitute tee vlue into te fomul fo te ufce e of te cn. S = 46.5; = 1.375: S = 2p 2 + 2p 46.5 = 2p p Te Model
6 78 CHAPTER 1 Line Eqution nd Inequlitie Wok Smt Round-off eo occu en deciml e continully ounded duing te coue of olving polem. Te moe time e ound, te moe inccute te eult my e. So, do not do ny itmetic until te lt tep. Figue 8 Step 4: Solve Solve fo. We ill not compute ny of te vlue until te lt clcultion. Ti i done to void ound-off eo. Sutct 2p fom ot ide: 46.5 = 2p p p = 2p Divide ot ide y p p : = 2p We ill ue clculto to evlute ti expeion. Figue 8 o te output fom TI-84 Plu gping clculto. Rounded to to deciml plce, e otin = 4.01 ince. Step 5: Ceck Te ufce e S of te cn it eigt 4.01 ince i S = 2p 2 + 2p = 2p p = 46.5 que ince. Step 6: Ane te Quetion Te eigt of te cn i 4.01 ince ounded to to deciml plce. 16. A cn of pece ufce e of 51.8 que ince. Te ufce e S of igt cicul cylinde i S = 2p 2 + 2p, ee i te diu of te cn nd i te eigt of te cn. Find te eigt of cn of pece if it diu i 1.5 ince. Round you ne to to deciml plce. 1.3 EXERCISES e te tt follo ec EXAMPLE Building Skill In Polem 17 20, tnlte te vel deciption into mtemticl fomul. 17. Foce F equl te poduct of m m nd cceletion. 18. Te e A of tingle i one-lf te poduct of it e nd it eigt. 19. Te volume V of pee i fou-tid te poduct of te nume p nd te cue of it diu. 20. Te evenue R of elling compute i $800 time te nume of compute old x. In Polem 21 32, olve te fomul fo te indicted vile. See Ojective Unifom Motion Solve d = t fo. 22. Diect Vition Solve y = kx fo k. 23. Alge Solve y - y 1 = m1x - x 1 2 fo m. 24. Alge Solve y = mx + fo m. 25. Sttitic Solve Z = x - m fo x. 26. Sttitic Solve E = Z # fo 1n. 1n 27. Neton L of Gvittion Solve fo m Sequence Solve S - S = - 5 fo S. 29. Finnce Solve A = P + Pt fo P. 30. Benoulli Eqution Solve p v2 + gy = fo. F = G m 1m 2 2
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