HFCC Math Lab Intermediate Algebra - 13 SOLVING RATE-TIME-DISTANCE PROBLEMS

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1 HFCC Mah Lab Inemeiae Algeba - 3 SOLVING RATE-TIME-DISTANCE PROBLEMS The vaiables involve in a moion poblem ae isance (), ae (), an ime (). These vaiables ae elae by he equaion, which can be solve fo any one of he hee vaiables,, o. Oinaily, one of hese hee foms of he equaion will be use in seing up a moion poblem. The fis sep in any applicaion poblem is o ea an eea he saemen of he poblem unil i is fully unesoo. In aiion, in a moion poblem all he imes given shoul be in he same unis, an all he isances shoul be in he same unis. The ae unis mus be compaible wih he ime an isance unis. Fo example, if he aes ae in miles/hou, hen he isances mus be in miles an he imes mus be in hous. In almos all moion poblems he aa can be place in a able in he foma shown hee: A B A an B may be caegoies such as wo iniviuals, wo vehicles, o wo ypes of eain. Seps o a Soluion Define he vaiable (he unknown). This will involve one of he columns of he able. Inouce any given aa ino he able. Ofen his will complee anohe column. Use,, o o complee he able. Wie he equaion an solve. Fo his sep each poblem is iffeen. We geneally look fo some saemen of elaionship no ye use in he seup involving ime, ae, o isance. Ex : A man walks fom poin A o poin B a a pace of miles pe hou. He euns a he ae of 4 miles pe hou. If he oal ime fo his oun ip is hous, hen wha is he isance fom A 4 A o B B o A o B? We le epesen he isance beween A an B, wih he fis ow of he able epesening his iniial ip an he secon ow epesening his eun ip: Revise 04/0

2 We now inouce he given aa. We ae given boh aes. (We omi he unis, which ae compaible, fom he able.) A o B B o A 4 Since, we now complee he able. A o B B o A 4 4 We now wie an equaion. In his case he equaion comes fom he saemen in he poblem 9 ha he oal ime is hous. Thus we have 4 4 (ime fom A o B) + (ime fom B o A) = (Muliply each sie by he LCD, 0.) 9 4 The isance beween he poins is hus miles. Ex : Two planes leave an aipo a he same ime an avel in opposie iecions. If one plane aveages 440 miles pe hou an he ohe 60 miles pe hou, hen in how long will he planes be 00 miles apa? We le be he ime of avel, since ha is wha we ae ying o fin. Each ow epesens he ae, ime, an isance of one of he planes. Plane Revise 04/0

3 Plane The aes ae given, so we ene hose ino he able: Plane 440 Plane 60 We use he elaion o complee he able: Plane Plane The equaion comes fom he fac ha when he planes ae 00 miles apa, he isances avele by he planes will oal 00 miles: The ime equie is hus. hous. Ex 3: Two planes leave an aipo a he same ime an avel in he same iecion. If one plane aveages 440 miles pe hou an he ohe 60 miles pe hou, hen in how long will he planes be 00 miles apa? We noe ha he able o be use fo his example will be he same as he able use in Example : Plane Plane Ou equaion changes, hough: (fase plane s isance) (slowe plane s isance) = 00 miles The ime equie woul hus have been 0 6 hous, o 0 hous 0 minues. Revise 04/0 3

4 Ex 4: A ca was iven 80 miles. If i ha gone miles pe hou fase, he ip woul have been shoe by an hou. A spee was he ca iven? We will le epesen he ae a which he ca was acually iven. On he hypoheical ip, he ca was iven a miles pe hou. Acual Hypoheical We ae given he isance he ca was iven. The isances woul be he same fo he acual an he hypoheical ips: Acual 80 Hypoheical 80 We use he elaion o complee he able: Acual Hypoheical The equaion is eive fom he saemen ha he ime fo he hypoheical ip is hou less han he ime fo he acual ip: (ime fo acual ip) (ime fo hypoheical ip) (Muliply each sie by he LCD,.) (Faco. If he quaaic polynomial wee no 40 o 3 facoable, we woul use he quaaic fomula.) In his applicaion, 40 is meaningless. Revise 04/0 4

5 Execises The spee of he ca on he (acual) ip was hus 3 miles pe hou. Soluions o o-numbee execises an answes o even-numbee execises appea a he en.. Two aiplanes pass each ohe going in opposie iecions. One avels a a spee of 70 miles pe hou; he ohe avels a 0 miles pe hou. How much ime elapses afe hey pass each ohe befoe hey ae 00 miles apa?. A pai of hikes, 4 miles apa, begin a he same ime o hike owa each ohe. If one walks a a ae ha is mile/hou fase han he ohe, an if hey mee wo hous lae, hen how fas is he slowe hike walking? 3. One cyclis peals a a ae 6 miles/hou fase han he ohe. A hose spees one coves he same isance in hee hous ha he ohe coves in fou hous. Wha is he spee of he fase cyclis? 4. A 4:00p wo vehicles leave owns 470 miles apa. They avel owa each ohe, one a an aveage spee ha is 0 miles/hou fase han he ohe. If he auomobiles pass a 9:00p, hen wha is he aveage spee of he fase vehicle?. A mooboa avels 0 miles/hou in sill wae. If he boa avels 6 miles upseam in he same ime ha i akes o avel 98 miles ownseam, hen wha is he ae of he cuen? 6. A boa goes 0 km ownseam in he same ime ha i akes o go 30 km upseam. The spee of he seam is 3 km/h. Fin he spee of he boa in sill wae. 7. A noon a Pino ses ou weswa a a seay ae fom a poin X on a naow, one-lane oa. A :00p, a Pius moving weswa along he same oa eaches poin X an suenly acceleaes o a spee 80 miles/hou fase han ha of he Pino. A 3:00p hese cas mee. How fa fom he poin X has each avele a ha ime? 8. A speeing auomobile is aveling miles/hou, an a highway oope, miles behin, is chasing i a a spee of 70 miles/hou. How long will i ake he oope o oveake he speeing ca if hey boh mainain hese spees? 9. A ive aveages a seconay-oa spee of 40 miles/hou an a unpike spee of 70 miles/hou. If a ip of 60 miles ove hese oas ook 0 minues, hen how many miles wee iven on seconay oas? 0. A man bikes a a ae ha is 6 miles/hou fase han he ae a which he jogs. Afe 30 minues jogging an 90 minues biking, he man has covee miles. Ove wha isance was he man jogging? Revise 04/0

6 Answes an Soluions. Plane Plane 0 0 (is. avele by Plane afe hey pass) + (is. avele hen by Plane ) = hous. 3 miles/hou 3. Cyclis 3 3 Cyclis (isance avele by Cyclis ) = (isance avele by Cyclis ) miles/hou 4. 7 miles/hou. We noe ha when aveling upseam he boa will avel a is sill-wae spee minus he spee of he cuen (which woks agains i), an we noe ha when aveling ownseam he boa will avel a is sill-wae spee plus he spee of he cuen (which helps i along). Hee we ll nee a vaiable, say c, fo he spee of he cuen. Upseam 0 c 6 0 c 6 Downseam 0 c 98 0 c 98 (ime upseam) = (ime ownseam) c 0 c 6 0 c c 70 c 4. miles/hou 6. km/h Revise 04/0 6 c

7 7. One meho of soluion: Pius Pino 3 3 (ae a which Pius avels) = (ae a which Pino avels) + 80 miles/hou The cas mee 0 miles wes of poin X. An alenae soluion meho: Pius Pino 3 3 (isance avele by Pius) = (isance avele by Pino) We now know ha he ae a which he Pino avele is 40 miles/hou. Bu we ae aske fo he isance ha i an he Pius have avele fom he poin X. We subsiue 40 fo in eihe isance expession: 3 becomes The cas mee 0 miles wes of poin X. 8. h (= 8 minues) Revise 04/0 7

8 9. We nee compaible unis. Since he aes ae given in miles/hou, we conve 0 minues o hous. One meho of soluion: Seconay 40 Roas Tunpike (ime on seconay oas) + (ime on unpike) = (Muliply each sie by he LCD, 80.) Seconay Roas Thus 0 miles wee iven on seconay oas. An alenae soluion meho: Tunpike (isance on seconay oas) + (isance on unpike) = 60 miles hou 70 The ime spen on seconay oas is hus a half-hou. We ae aske fo he numbe of miles iven on seconay oas, hough. We subsiue he value of ino he expession epesening he isance iven on seconay oas: Revise 04/0 8

9 0. 3 miles 40 0 miles iven on seconay oas You can ge aiional insucion an pacice by going o he following websies: hp:// This websie gives eigh aiional examples. hp:// This websie gives an aiional example an fou pacice poblems. Revise 04/0 9