8.F Baery Charging Task Sam wans o ake his MP3 player and his video game player on a car rip. An hour before hey plan o leave, he realized ha he forgo o charge he baeries las nigh. A ha poin, he plugged in boh devices so hey can charge as long as possible before hey leave. Sam knows ha his MP3 player has 40% of is baery life lef and ha he baery charges by an addiional 12 percenage poins every 15 minues. His video game player is new, so Sam doesn know how fas i is charging bu he recorded he baery charge for he firs 30 minues afer he plugged i in. ime charging (minues) 0 10 20 30 video game player baery charge (%) 20 32 44 56 a. If Sam s family leaves as planned, wha percen of he baery will be charged for each of he wo devices when hey leave? b. How much ime would Sam need o charge he baery 100% on boh devices? IM Commenary This ask has sudens engaging in a simple modeling exercise, aking verbal and numerical descripions of baery life as a funcion of ime and wriing down linear models for hese quaniies. To draw conclusions abou he quaniies, sudens have o find a common way of describing hem. There are hree soluion echniques
presened below: a. Finding equaions for boh funcions. b. Using ables of values. c. Using graphs. There are also ample opporuniies o alk abou he role of modeling here, ouching on mahemaical pracice sandard MP4. How reasonable is i ha he oupu unis are repored as percens? Does he model hold for all ime? In paricular, noe ha he model predics ha he percen charged grows linearly for all ime, even beyond 100%! If he ask is done in small groups, differen groups would likely use differen represenaions in heir soluions. Having groups presen heir answers could lead o a rich discussion on connecing differen represenaions of funcions. Soluions Soluion: Finding and using equaions a. The baery charge of boh devices can be modeled wih linear funcions. The wording describing he MP3 player suggess a linear funcion since i uses a consan rae of change. The able of values for he video game player shows a consan rae of change for he firs 30 minues. I is a reasonable assumpion ha he baery will coninue o charge a he same rae. However, i is an assumpion on our par. (Anoher possibiliy would be ha as he baery charge approaches 100%, he rae of change decreases, bu ha would be much harder o model.) The MP3 player charges a a rae of 12 percenage poins every 15 minues, which is equal o 0.8 percenage poins per minue. If we le y be baery charge of he device (in percenage poins) we have: where is measured in minues. y = 0.8 + 40, We know ha he video game player is iniially 20% charged and from he able we see ha he charge increases by an addiional 12 percenage poins every 10 minues, or
1.2 percenage poins per minue. So for his funcion we ge: y = 1.2 + 20. Sam s family is planning o leave he house 60 minues afer Sam sared charging his devices. We are looking for he charge when : MP3 player: video game player: y = 0.8 60 + 40 = 88 % charged y = 1.2 60 + 20 = 92 = 60 % charged b. To answer his quesion, we need o find he values of for which each funcion has oupu value 100. 100 = 0.8 + 40 = 75 MP3 player: Solving for we have, minues. 100 = 1.2 + 20 = 67 video game player: Solving for we have minues. So if Sam s family could wai jus 15 more minues, Sam could have boh devices fully charged for he car rip. Soluion: Using ables a. Since he video game player s baery charge is given in a able, we can exend he able and see wha value i will give afer 60 minues. Noe ha he rae of change of he daa in he able is consan: For every 10 minues he charge increases by 12 percenage poins. Assuming ha his paern coninues, we have: ime charging (minues) 0 10 20 30 40 50 60 video game player baery charge (%) 20 32 44 56 68 80 92 We can make a similar able for he MP3 player: ime charging (minues) 0 15 30 45 60
MP3 player baery charge (%) 40 52 64 76 88 So afer 60 minues, he MP3 player s baery would be 88% charged and he video game player will be 92% charged. b. We can see from he able above ha he MP3 player would be fully charged in anoher 15 minues, we jus have o add one more column o he able o find ha answer. The video game player will need less han 10 minues o fully charge, since we are only missing 8 percenage poins afer 60 minues. To be exac, using he rae of increase, we will need 2/3 of 10 minues, which is jus under 7 minues. Soluion: Using graphs a. Wih he given informaion, i is quie sraigh-forward o graph he funcions for boh devices. For he MP3 player we have a saring value (i.e. verical inercep) of 40% and a rae of change (i.e. slope) of 12/15 = 0.8 percenage poins per minue. For he video game player we have a saring value of 20% and he rae of change for he daa in he able is consan a 12/10 = 1.2 percenage poins per minue. Below are he wo graphs. We can esimae from he graph ha afer 60 minues he MP3 player has a baery
charge of jus under 90% and he video game player has a baery charge of jus over 90%. Zooming in on a graphing calculaor or oher graphing device would give us beer esimaes. b. To find ou how long i will ake unil boh baeries are fully charged, we need o find values of for which he oupu value is 100% for boh funcions. From he graph we see ha he MP3 player will ake he longes o charge and i will ake abou 75 minues oal. So if Sam s family can wai an exra 15 minues before hey leave, Sam would have boh devices fully charged. 8.F Baery Charging is licensed by under a Creaive Commons Aribuion-NonCommercial-ShareAlike 4.0 Inernaional License