Sectn : Detaled Slutns f Wrd Prblems Unt : Slvng Wrd Prblems by Mdelng wth Frmulas Example : The factry nvce fr a mnvan shws that the dealer pad $,5 fr the vehcle. If the stcker prce f the van s $5,, hw much ver factry nvce s the stcker prce? Analyss: Ths s a straght-frward subtractn prblem and des nt nvlve a frmula. Subtract what the dealer pad frm the stcker prce, $5, $,5 t btan $659 as the markup. Fat cntans 9 calres per gram and bth carbs and prten cntan calres per gram. A frmula whch relates calres t fat, prten, and carbhydrates s: Cal = 9F + P + C where Cal s the number f calres, F s the number f grams f fat, P s the number f grams f prten, and C s the number f grams f Example : A servng (/ cup) f splt pea sup cntans g f fat, 9g f carbhydrates, and g f prten. Hw many calres are n ths servng f sup? Analyss: The relevant frmula fr ths questn s Cal = 9F + P + C. Substtute the knwn quanttes nt the frmula t btan Cal = 9() + () + (9). Nw perfrm the multplcatns fllwed by the addtns t btan Cal = 78. The cnclusn s that the servng f pea sup cntans 78 calres. Example : A servng ( tbsp.) f peanut butter cntans 5 calres, 5g f carbhydrates, and 6g f fat. Hw many grams f prten are n ths servng? Analyss: The relevant frmula fr ths questn s Cal = 9F + P + C. Substtute the knwn quanttes nt the frmula t btan 5 = 9(6) + P + (5). The slutn t ths equatn wll answer the questn. 5 = 9(6) + P + (5) 76 = P P = 9 The servng f peanut butter cntans 9 grams f fat.
Example : A servng ( z. -- abut fres) f ven baked French fres cntans 6 calres, g f prten, and 7g f fat. Hw many grams f carbhydrates are n ths servng? Analyss: The relevant frmula fr ths questn s Cal = 9F + P + C. Substtute the knwn quanttes nt the frmula t btan 6 = 9(7) + () + C. The slutn t ths equatn wll answer the questn. 6 = 9(7) + () + C 9 = C C = The servng f French fres cntans grams f carbhydrates. Ohm s Law Ohm's Law establshes the relatnshp between vltage, current, and resstance n an deal cnductr. Ths relatnshp s usually stated as: V = IR Example 5: Where V s vltage, I s current, and R s resstance. What s the resstance n a drect current electrc crcut wth vltage f 9 vlts and current f.5 Amps? Analyss: The relevant frmula fr ths questn Ohm s Law V = IR. Substtute the knwn values nt the frmula t btan 9 =.5R. The slutn f ths equatn wll answer the questn. 9 =.5R R = 8 The resstance n ths crcut s 8 hms. Example 6: What s the vltage n a drect current electrcal crcut wth current f.75 Amps and resstance f hms? Analyss: The relevant frmula fr ths questn s Ohm s Law V = IR. Substtute the knwn values nt the frmula t btan V =.75(). The slutn f ths equatn wll answer the questn. Perfrm the multplcatn t btan V = 8 vlts The vltage n ths crcut s 8 vlts.
Resstrs n Parallel The ttal resstance n an electrc crcut wth tw resstr cnnected n parallel s gven by the frmula: R R R Example 7: What s the current n a drect current electrcal crcut wth resstance f 5 hms and vltage 6 vlts? Analyss: The relevant frmula fr ths questn Ohm s Law V = IR. Substtute the knwn values nt the frmula t btan 6 = 5I. The slutn f ths equatn wll answer the questn. 6 = 5I I =. Amps There s. Amps f current n ths crcut. Example 8: If an 8 hm and a hm resstr are cnnected n parallel, what s the ttal resstance? Analyss: The relevant frmula fr ths questn s. R R R Substtute the knwn values nt the frmula t btan. The slutn f ths equatn R 8 wll answer the questn. R 8 7 R 56 56 56 If tw numbers are equal, then ther recprcals are equal. Therefre R = 56 The ttal resstance n ths crcut s 56 hms. Ths wuld nrmally be wrtten as 5.9 hms. Example 9: It s requred that the ttal resstance f a crcut be 6 hms. If ths crcut cntans a 5 hm resstr and a varable resstr cnnected n parallel, what shuld the settng be fr the varable resstr?
Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt R R R the frmula t btan. The slutn f ths equatn wll answer the questn. 6 5 R 6 5 R 5 R 6 5 R The varable resstr shuld be adjusted t hms. Example : The ttal resstance f a crcut cntanng tw resstrs n parallel s 5 hms. The resstance f ne resstr s twce that f the ther resstr. What are the resstances f the tw resstrs? Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt R R R the frmula t btan. Ntce we made use f the fact that the resstance f ne 5 R R resstr s twce the resstance f the ther. The slutn f ths equatn wll answer the questn. 5 R R 5 R R R 5 R R 5 5.5 One resstr has a resstance f.5 hms and the ther has a resstance f 5 hms. Example : A hm and a 5 hm resstr are cnnected t a vlt battery. If the resstrs are cnnected n parallel, what s the current n the crcut? Analyss: The frmula wll permt us t calculate the ttal resstance n the crcut R R R and then we can use Ohm s Law V = IR t cmpute the current. Substtute the knwn resstr values nt t btan.the slutn f ths equatn wll yeld the ttal R R R R 5 resstance fr the crcut. The current flwng n ths crcut wll be 7. Amps.
R 5 R 5 R Substtute ths value and the knwn vltage nt Ohm s Law V = IR t btan I. The slutn t ths equatn wll be the current n the crcut I I 6 () 7. 5
Thn Lenses If a lumnus bject s placed at a dstance greater than the fcal length away frm a cnvex lens, then t wll frm an nverted real mage n the ppste sde f the lens. The bject pstn, mage pstn and fcal length f the lens are related by the lens equatn. f d d where f s fcal length, d s the bject dstance, and d s the mage Example : If the mage dstance s centmeters and the bject dstance s centmeters, what s the fcal length f the lens? Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt d the frmula t btan. The slutn f ths equatn wll answer the questn. f f 7 f f 7 The fcal length f the lens s cm r apprxmately.7cm 7 Example : If the fcal length f a lens s 6 centmeters and the mage s centmeters frm the lens, hw far s the bject frm the lens? Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt d the frmula t btan. The slutn f ths equatn wll answer the questn. 6 d 6 d d 6 d The bject dstance s cm.
Example : If the fcal length f a lens s centmeters and the bject s 5 centmeters frm the lens, hw far s the mage frm the lens? Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt d the frmula t btan. The slutn f ths equatn wll answer the questn. 5 d 5 d 5 d 5 ()(5) ()(5) 6 d 6 The mage dstance s 6 cm r apprxmately 5.5cm. Example 5: If the fcal length f a lens s centmeters and the mage dstance s ne-thrd f the bject dstance, what s the bject dstance and what s the mage dstance. Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt d the frmula t btan. Nte we have made use f the statement that the mage d d dstance s ne-thrd the bject dstance. The slutn f ths equatn wll answer the questn. d d d d d d d d d 6 The mage dstance s 6cm and the bject dstance s 8cm. Example 6: If the bject dstance s twce the mage dstance, what s the fcal length f the lens? Analyss: The relevant frmula fr ths questn s. Substtute the knwn values nt d the frmula t btan. Nte we have made use f the statement that the mage dstance f d d s ne-half the bject dstance. The slutn f ths equatn wll answer the questn.
f d d f d f d The fcal length s the mage dstance. It prbably makes mre sense t state that the mage dstance s the fcal length and the bject dstance s tmes the fcal length. Example 7: A cmpany has establshed that the number f bjects a new emplyee can package per day s gven by the equatn N 5t where N s the number f bjects t packaged per day and t s the number f days f n-the-jb tranng. Jhn packages 7 bjects n a day. Assumng Jhn s an average new hre and fts the mdel, hw many days has he been n tranng? Analyss: The relevant frmula fr ths questn s N 5t. Substtute the knwn value f N nt t the frmula t btan7 5t. The slutn f ths equatn wll answer the questn. t 7 5t t 7(t ) 5t t 7 The emplyee s perfrmng at the rate f smene wth 7 days f tranng. Fallng Bdy If an bject s fallng and s subject nly t the frce f gravty, then ts pstn abve the earth s : y y 6t Where y s the dstance abve the earth, y s the ntal heght abve the Example 8: A ball s drpped frm the empre state buldng. The empre state buldng s 5 feet tall. Hw lng wll t take the ball t ht the grund?
Analyss: The relevant frmula fr ths questn s y y 6t. Substtute the heght f 5 feet fr y. Substtute fr y because when t hts the grund ts dstance abve the earth s feet. Ths yelds the equatn 5 6t. The slutn t ths equatn wll answer the questn. 5 6t 6t 5 t 5 9.875 6 t 9.875 9.5 Slvng the equatn yelds a pstve and a negatve value. We may dscard the negatve value because t s a tme after release f the ball and must therefre be pstve. It wll take the ball abut 9.5 secnds t ht the grund Example 9: Lenard (Bg Bang Thery) drps an bject dwn the elevatr shaft and cmments that t tk. secnds t ht the bttm f the shaft. Frm what heght abve grund was the bject drpped? Analyss: The relevant frmula fr ths questn s y y 6t. Substtute the tme. secnds fr t. Substtute fr y because when t hts the grund ts dstance abve the earth s feet. Ths yelds the equatn y 6(.). The slutn t ths equatn wll answer the questn. y 6(.) y 6(.) 6.8 The bject was drpped frm a heght f 6.8 feet. Example : A ball s drpped frm a heght f feet. What s ts heght abve grund secnds after t was drpped? Analyss: The relevant frmula fr ths questn s y y 6t. Substtute the tme secnds fr t and fr y. Ths yelds the equatn y 6(). The slutn t ths equatn wll answer the questn. y 6() y 96 6 The ball s 6 feet abve grund after secnds.