Table Of Content

SOLVED PROBLEMS IN CHAPTER 14 Related Rates. mo wa us “tne wp ofa 264s dae, ening grin et! wailing dove wa tae cate of oo Pe ean Fee ress tne htt of he gossiping along the arob0d when tbe bottom af he ladders? fest sway fo the hase ofthe wall? ¥ ve tt Lethe tte distance of the top oft adder from the gronnd, ad ets the dane i the bottom of teat Jom hs bse OF ths wal (Fg. 1-1). By he Pythagorean tbsorem, 3? + y'= (25). Dieeria inteepeu wume sar Byrsay. Dy 2 x1 Dz+ ¥- 129 =0. The gienifomation tls vt eS oot por ascont. (sac che baer i sing down Oe wal, is daaessog, and, thersfone Breive nebatoe) "Wheo n=? subtaton nF)" = (O5F yields 7° —5T8 y— 2H, Sobel Moola ge Dyety-Dynt soli 7-De-r24-(=1)~0, Dpe~ # fest pe stood ‘Acptcal eo of ius 10 fet sing led wi whoa atthe ate of 1 ie et per minute Hom et EEueninot the when imtoadog? ihe vobme of jliter 6 wrth, were 5 Ws radio nl sits hes 1 LecVbe ie vohuve of wheat atime aed lr he she depth nthe wheat inthe tok. Then, ¥= oft Sob elte: Bik, Dalwe ae goon at Dy'=stocube fet per minute. Mence, 334~ 100m, Bi Fics) Thee appontnnte why 34, tben De 1, Th he depo The went nea tho vate of eat foc pe multe ‘A oot gl walking tywurd «2040 mot othe ae of 6 feet per seco. How fst se dp ot Shadow (at by help) ning? My ma fcc be the tance ofthe gi fom he base of he post, age Bed biaoce of che ot he ha fon ite mutt of tr pom (Fig 42), DABE W sila © APEC. Heowe, AB'DE=2(y 20. 3 Heinys beg bey inde Tienes, 3D —4: Dz, Bat we ae wid tat Dix Wa yeetaand sie ah is wating twa the bane, # is daresing, aod 2 Ruegatse,) $0, 2 UD ny 8. Thaw Gc tp ofthe shadow is moving at herale ofS fet yersecond tonal base of po Under the ssi: contin sm Probe 4.3, haw Fa athe ent of te g's shadow chung? 1 ie thesume notation win Problem 143, Let Phe the Teng of Reeshadow, "Theo se 7 6 PELATEO RATES 2 eg Dez Dy~DamC 8 6)~—-2 Thus he length of de shaw is deteaing he vale 2 feet pot ‘eoend ‘A rocket is shot vercalyupmata wih iii eli Of 4 foes. beiht 5 after seconde it S=4001—16/ How iat vibe dance ebungng fom he rocet oan nerves on he grou 800 fst ay from the ucehing ste, wen he retell ing 900i 40] fat thine the Ground 7 Fig 49 4, Lat be te tans fom the sock tate aber, a show io Fig. 143. Ty the Pbagorcanaeorn, ae TOMI Hepes, Be Da n2e Di a Daze: Da. When 22M, (OY (OO), ee fo0:30~ Sam. Sige $A “16, hea p= 2400, 24N0—aM = H6e. YF —35E 1-0, G10) 15)~0, “So, on the wa up, the rvket at 400 et when = 1. ag, z= 400" 3s. Su, Whee IU, Dg = A030 1D= 0. SUbsRCUCAgN u-Dae—s 22, we obtin 300: Diu 2e Dykml. So the disnote oat the oske othe steve i acttanng at the ste of feck per second we aa ‘aml fant inthe shape of cance ny emptind of Bd ate rte uf 2 bic oncinetew per send. Tae Sigh of We tunnel i 20 ceoimeters and the rads of De op #6 centimeters How fat heli level ‘topping when the level stands Sceatonele sone the vertex ofthe sine? (Rerwember fat he me ok cone ger) s Fg The id inthe tame fomms a cane with acs, Bight ant volume ¥_ By sac langles, 4 = 42D, SS. Vo barks sa(hiShk~ deni Heose, DV= dre". Oj, We ue ahem that DY" 12, since the tearing a Re vate ot 12 abi cntinctcs yer sere. Hetes. 12= het? Bi HM= Hh Dh. When AS, “We 26 Oh, Df iSley oe approninatly ~3 2 eaten person, Hoc, the Mi ets droping atthe ate of about 2 entice ot Send, ‘A balloon is being infuted by pucaped aly atthe race cf etic inches per igvml, How fasts the diameter ofthe aloo ieeasing when he rai it isk? so mals Py was CHAPTER 14 J yada So, DY = ser? Dy. We ate told te OV=2, So, 224e7"~De, When rd, 2 dab bn Destin La d be the damete. Teen, d=2r, Del 22 Dy =2-Cin)= dle 1.27, ‘So. the ete is inscaing at the ete of abot 1.2 inches per ses, ‘0a tons an uncapped wel io tbe oean i aang ona in she form of eta ln onthe ste ofthe Oa om tae orn the ale creaaap oS at often ot minute, bow faethe area he owing wen the rai i 100 eters Fotve aca Az we. $0, DATInr Dy. We a given that Dyr=2 Hess, when r=100, Anda eid: 2=A00e, wiih ie abou 186 mri, “Te lens of a restngle of costar ar 800 square milimeters is mctsing the rate of tle por Fond eatin tbe ho the secant te omen the with creasing othe rate of 0.5 minor Per teoon? Tote sce WO=Ew, Dilorenotng, O= C-Dw tae Dye We we goon tat BERK Su, O~ ED lte, When Byrn OS, UE DSE re, wm O80, Bu C-80w, So, dw —O5IEUD = foe, 7100, w= We, Cader the ce conditions an Pct 4.9, bow sti th agonal ofthe rctangle hanging when the width Mast TAs in the xluioo of Problem 149, O=¢-Dy +4 Let u be the cians. Then t= wt LD wt keDe, a Daw Rw l CDz. When w= 2, f= 0in A, Soba io Bee bprae!” O=We bet Me Dye"? When w= 20, a= 07 | (40)? =200, a 208, Sonate a Duen we Bw t-,e, MVE Da 20-(-2)4 40-42 1D, P= tV5/5= 260mm! Ayucice oovesin thebyperbote x! 18)°=9 Sn coch any hat ly }enontinale neces a onstant rate cB unis pecsecind, How fot ssa-coordinnechangig when x= Fo 26:Da—My- By =O, ¢-DyW Why Dor, We ae gre that BY =D. Hence, Dir = 1899 — Noop When en 3, QP aU ek Aha ged, yo S2 Salmi in DEW Hy, Seen BK, Daye #36 unis per second, “Apskject moves slong graph of y-~fl). Atacenaia pint, the pst the carve sd the xenon ‘Tetwnfeee dearesing atthe cat of 3 ola per socoad. Wat post, fast the y-coordite of th jet clanging? Fo y= fa) By the iin sul, Dy =F'ta}-Da. Since (0) tbe lope E and Bi }-3)~ fw por secon Li the rads of a spheres necaiag at tbe sant rare ofS alimeters per si00d, Bow Git sth volume hanging when to auroras bar 5 10 sgooe soineters? F vitor tense, DY=4ne-Oor We ace given the Se7210, 1y oso’ far3 Wen svat isthe min uf an expending se ta moment wen be ete eange os area cued ic os Tang athe rte af change of fe mis? Po dwce, Hence, BAS2er Be. Wea PAT De, BAga2er- Dy, t= 1k paride anes ong the cure y~2s7—3e? +4. AL 8 corain momar, when x2, the parle’ ‘ehordinate i fosecong tthe ele 060. oil pe sesond. How fac He ysooroate changing a at B dy 60s Da~Ge- Dy~be-Dale-I). When 2, Be 1200511) ~ ale par seoone, 5. So at that omer, Dy ‘A ple fying stall tothe youn ata eight of leoeters pases overs rar ton Rig, 145). Ashort ‘Gn te, hc rndar equpmettvevels that the tence between te plac and the dation 5 ere an that {he datane ewoeo the plane and heaton ripening ce of°0 Wooeterrpec tow. At shat moment, how fs the ple noing boro? RELATED RATES 2 91 Fast Ws bt bodzotl tne of esse oe te pit itor Rs lt abe he ace fence th pte age sation Thon Waa) So, 2e-Dyra de Da, w-Dane by Siren "eats, (Spee Ge ves and we ay aaa old Gat Oyen SO Subtang wt 8-2 Biba, “S3i9=3-DD, Pg Hs Rig 18s ‘Anos paies Sed toy a.m. Reading due wert Tec pe on. Another bat ptt the ste buoy at arm heading due norh tS ves er hour. flaw Ete ditance tween he ost changing a 11! 30 # Refer co Fig. 14-5, Let the time be measured In bous afer 9a. Lets hee mba of estat the fcc bat x weet ofthe buoy a ave, and ety Bethe nab asa the seta vb te Wy a time tLet v be the danse between the bose atte 1. For any time 12), wheat ge Then 2 Duca Bs-De+2y- Dav we Dune Day Dy. We we Gren that Daw3 and BYS. See B Gym 3y—57, AP U-iham the fst Beat has traveled 2! bows at 4 sles per bow; 4, == Solty, the stcond bout bag Gavelled at Smiles pr bou for 1 haut sige gale the ys a0, 9 =F Aig w= GBP + (BY =, aa isivE Substiningia w-Da= 3x35, OSB) Bund: Pb Fo een, Dyer ev Fo 8 ces per tour ‘ater i posing itn an iver cane a thereof 3. enbie meter pot anate, The eight of he cre £0 refers ad the rebar fis Ore mrtere Hi ac then eel sing he te wae and 33 mete Fig ter 1 Leto Sethe vel ofthe waier above the base. nd ets be the af he cde hat forms the waka of tbe wate Ley 210. Thoa >is the heigh of the coocahape region above the water (ee Pg, 127), 8, the volune of thal come ie 'Y,= Jury. The tl value ef che cme sonainet Y= s¥ 10 28oy!3. Thos, the total whine of (he water ie V=U,—¥,~20e!s— ar y!3, By emir tare, WS= yin pode Say Vm2OwFd~ or Pla 2a ae ence, Ve Bere We areniventiat” DY=314 So, 362 —2or Be Thus, Dr=—216i2ar" When hewaterstande Toctenintoecone, 75, y-10-24u25 Hy =1T So Dye —AWENLDSN. Dy= 2ippas Stdlm( 2S, Dywm - Diy 3.88 /a( 25 = U8 min Aputidlermoveraongihecuve yn 2s. Atwhacpoints)on the cuve ae the. dp cordinter ofthe omle cong at tie ame ate i Bo ngate nes? ny=ns2er2, When Dy= Dx, WetIe1, tem 9 2 CHAPTER 14 14.20 A oat itbeng pulled into a dock by arope thc panes Uynntnasing on thebaw ofthetont, The docks fet aber tha te bow eng, How fare the tot appreting tke dock hon the Tenth of rope between he doe So he bot i 10 fot, the ope i being pled in De cate of eo pee second? Bip ts # Levebe rhe honzonal dstanee fom ie bow dag tothe dock, ult he lent theca bets Musk and the Bante beh, ie nat HPS fe- Dante Dz we Dews Oe. We ace tld Ua Dan 'S So dena Da. Ween v21Q, C28, 2-8 Hen, “Sled De, Ben S80 the Bout & appoeshing he dock tthe rae of 55 MA ise yng abit, whiiaatabeight of WDfeu!, The wind cortog the ttc ortcasly as fom he gl Sciapesd of feet pe steond. How fast mit the We sri bellu we the Sung i 180 fet La? J Lecebe ec horlaonal distance oftbe St frome poige rest over the guid at 2Dfeet. Letabette feng ofthe hte sug fas the gid foe te, Toen "7" {IBN So, Da Dem 2s Da, ae Da = MBs, We a told Gat Dwi. Henge, # Dam iis. When «150, 9° 8100, -¢~90. "So 10-Due= 900, Dye 6M i ‘Waa2 A roctugula ough feet lang. feet across the tops el ee sep, Tater Town at 3a of 24M how fasts he surace cng when the water is 10 een 1 cess bere depth af water, The he waters rectal ssh yf ensues, 2 se Hewes hs olan Vibe, So DVI Dae We ae told tat YT $0, 218 Dp Hence, r= 1433 ladder 20 eet long ean aun 9 bone, Rnd the ent 3 shih het of the nde moving own the foot ofthe wee fet ay fom che bow at ising lo he grounds fom The howe ea ot fre per sceand? Fara te ne ataise ot me for af me adder from he bat ofthe Rou, ese ye thence of he the tia fom the quel Than at GO). 50, 34-Da ede Dy =O, xD} De~d I We ae wld that z= and Dea? Wheo x=12, y°256, 7 — 160 Subeiting to «Dz pDy ne WDA Dy 0. Diya of. So the ler ting dow the wall wie rate of 1.8 i 1424 In Prouem 14.28, now ft i the angle « Decwoen the ladder andthe groin’ changing 3 the given mace POyny De WH We2 o ae a Alo, ana mye HA. Se, ata nt tuo en rife g. Ths, Fe Dan. Denn ence, the angle decresing it the me aff aian pr Secon Fouanasyis, So, by the ohain ns, acta, N25 A tin unin at 1 2.4m eve em 45 mile yor haut. whl anosher stating at non fom the sme oi I ‘ate South at nites pe hour lst Be tance beteeoa Dees Iczesig a3 Ban? F Leche ime be mented io hours sting am. Let eli tbe estan that the Ct ain ens sting pot and ety be the sage at the recon rin esouthef the aring int Late the isan etween the tains “Thon waar yy Du Dende Beet 2y- Dy, w Bene Dat yD Wen fo mat Die =A, "Sou. Due=4Se-+ tly. ALS pm the fre Li hos bot abel Frc hmiceat Sith 0d, tberfore, 180; feed main heen raven for Tore i ae at ae RELATED RATES 2 93 Uheseore, 9~ 181. Them, w= (180) 180} sem MOVE thas, IBDVE- Dy = 45-180 + €O-0H0, Darn VE 2m [A ihn cae pot pele 80 ot ah. ball is deopped fom the ste beg (0) fram pint 3 ect From the ight. ASsooog tht Bo ball alls scoring othe ew = 16P, ow fat the shadow of he bald snoring alg tbe ground one sso0d ater? F soe Rp 1469. Lets be the dane ofthe shadow ofthe al fom the bane he ighipole.Lety be the beght ofthe all shove the ground. Sy vil Imeles, 940" (2 Myre. But p= BD 16e So Vje1= COA). Difteaaiag, ZrO) Bae Whee fate 1 P= 1dr 2 0 Svbsitwing in —21-@0ee")-Dix, By, ence, the sada a svi 200s ig 0 ‘Shy A i038 mites est of point Oandsnaving wee at Oma pechout. Ship Ais 60 mils south Gand mowing Iori a 18 mies er ou. Arete approaching or weparting afer Uhr and a hal ate? 4 Let te point 0 be the ovgn os ordinate syste, wlth 4 monng on the Smad ad # movngon the 285 (Wig. 410) Since A begga et 24S and st moving tothe lett enh, is poaion ewe, de potion of Bis y~—00-4181, Let be the dntance between A at Thea Tdurar Dee%y-Dy, WDurzDxty:Dy. hace Dee % wd Ayeise SMe 5p. When P80, y= 1 m=—§, ya) t Sends, ESE AF = SHED), SVEB Satmiwting io ua 20415). WEDa~ SIE, Burm “lISVEE~ 18 Snes Ge Live ota ont, he etance Derwent vps i gtng sls, a soupy ih, ‘oder the same hypotbeses a a Probl 14.27, when ate the sips mere enc other? 4 vihen the ships ste veanat cieh ther, tele dstanc a atoms a eltve moun, und, therefore. Dax, Subuititiog to ue Dae~ —Hx~ 13). Om Ae e. Dot sis Hh and po 4 Tee Se, D3-20(15~20) 7 19-= 18H), = hous, or appeonimately, i or 2a 35 mines ‘ote, tthe ate 10 cube fet per inute, s pouring ito a ey tera whose shape cme 6 feet dep and Fa ea Mat re CHAPTER 14 (fet ameter at the top, A the ne the ater 1 fet deep the water Tvl bere cb is Faces peraloute, How tet the voter sig U0? F. Leche the depo the wate, ad et be te rans ofthe wer surface ig. 112), The water's or Vo forth, By sna ities, V4 ALG, r= ah, U—Fethish~ Bebe So OV hak, Weroe wldthstohon A= ty DA i Hence, at lc emoment. Y= de l44yb}= 3a. Since thet whic te materi potting fn i she ate Teabags (10— 3a} Pei, An aitpleneia mcendiog ot sped of 40 kilometnxper hour slong ine cuking an angle of 6" withthe grou How tat athe atade ofthe plane change? eth ee tins oft ae ae eh uns of ptr ot oud ag Bh (Heid. Then MuraesPevI2 2H Vn. 2 Dp meters pee how Fig i Fg oS shadow eat on evel ground by gol 50 fe al engthening when the angle af clevaion of son ast an iedeerwing By sadien poe Rou? (Soe Fig 14.16) 4 Lecrteterengt ol esha ae S0', By estate, ee DCS) Da 4, tingly seem ioe Hass, Ds 25h A eetlvogfenovn i insted 3600 ft of «stag shone. The bean rn at drains pe ious faa dos its Beam smerp sous te shore ats nears point AP * ig 414 # Letxzethecinane rm 4 tothe print on theabure a bye beacon, an let abe tbe angle becween he fromthe ligthoane Bie Acind the beaton (Big. 4), ‘Thot tan 303000, e060" Dye = bP ease cd thet Djamdr. When the beaton Nie print oO, vaca 3, 80 bem del Byea 1h n= 2 “Two sides ofa langle are 15.0820 feet Lng, empestvely. Hoe at the hind ide neni hen thea co eetwonn the ye sce ie" and i nresing tte rate per avon? f Letrbehe tive. By the lof cosines, x= (157 +(2)'— 21S HAN cova, Hence, 24-0, GOsina Dye 3-Day Sond When 3 Giga t VER, anand 42125 +400 600-F FaovIS. Hence, 3VT- Dj HO-(V32)- (2390), BE= (oY Rs pret] as ey at was aaa RELATED RATES 2) 96 ‘The are of n expanding real a aeesing athe ene of 4 quae centimeters per sec. The length the cestngle alway egal to the square off wi Gin censnetes} At arte the ebgthaccaing st fhe intent ten Phe wih ie 2on? Bo Amt, and Cow So, Ame! Hane, DA % DAH Henie, B37 Dion 18 Dy When wad, WE Dm, Diem 20 Dae Heoie, B,222-1-4= sem ‘A sphecelsoowha is meling Gymmetinly) atthe sats of cubs contmetes per bout, Ho fi the lamer changing when is 2 comme Bo The wohume Veja So, Bymter-0p, We mre tok) that DY —t. Heme, ter 4a Dy Tas, “157° Der hen te dumetore 0 eentineters, toe cada r=, Hanes, 7 WO: Dy. Dy "00. Scr the meter d=3r, Dat=2 Dy =2(-DON)= 002, So, the dameter i doeoeting athe rate (002 centineter por hoe, A ttough i 10 for long so has a cos secon in he akope of an ego tcanle 2 feta ach side (58. C15). dPveater is being pumped atte rate of 2042 mn, hom fortis heme sel ig eh the Water 18 den? Fy. as 4 The water ia the cough vil havea cot sts hat can equoteraltiagle, say of heh andes In an eguleral tangle with stk 4. ¢=2h1V3. Hence he taveaeconalatea of the water PQhN3)- b= WAVE Thesefore, the wwluwe Vuk water 8 WHNG. 5, D,V= (RIVE) De We re‘old tint YET So, MS CON DA, VBI DA. When hh, Dh=vitiain a oot evaporate ats ne proporonal tits surface aces 47% show tts aie decade a consent J The wtume Ve der, So, DV =4er" Dr. Weateold that BY—K-4xt* for sme constant Denes, R= By. Sands being poor eno a conical ile atthe constant ae ofS) ebiefoe peraoute, Frans forces in the ‘dou ae nach ha he eight of Sep e alwaeeun tne radi ote, Hw at ie eight of he pe increasing when the sane 5 fet deep? F Tresclume = forth. Since hor, V= dah? DY= oh-Dh. Wesre wid oat se Waa DA. Whe RS, Mo r-25- DA, ‘At conan moment, 0 anmple of yo ebeyng Hepes lee p= constant, ecapst a wlume Vf 1000 enbie inches eta pressure p of 10 pounds pee ageane ich Ife gi blag eamprtied ate ae of 12 ube ines er minute, fd the te at ich the peu nema "te neaD he Ine alate oD cat inches 4B Since p¥=conmant, p-D.V+V:DipwO We are tld that DV= 12, a0 -12p+¥- Dip 0. Wen Y= F000 and p10, "Dp = 0.12 pred per square och pes mute ‘A Inder 20 fet long is eng again» wal 12 fot igh wi it op proving ave te wail (Fg. 1416) Dosim sbeig pled avay froma tbc wal she conan rte OFS Rea. Hot yy the high of be op of the later decreasing when the top ofthe ladder eases theo of he wall? 4 Lexybe the neigh of be top ofthe tr, fl be the Stane ofthe Rotor ofthe adder fro tbe yal nd let w bo the ubsanor from the bottom of the eddet co the top of the wall. Now, a =x C2) Be Dun2e- Dye, w-Danx-Da We med that Dyc=S, So, a" ByrnSe. Whoo the tp Be 96 0 CHAPTER 14 Inder reuhes the Cop of the wall, 220, ee (NE -(2)'=286, 2516 Mets, 20: Da $16 Dance By staier ingles, 9S12-220/0, yo 2ien, Dy=—QON)- Oem Hea= EA Rinin ‘Ts, the height of toe Isr deccaing a hereof 2.4 fot per mae A ess = neu at ree is Being poured its besisphericl bow of min 3 aches atthe rate of Levis inch pe secood. Mow fonts the wales evel ing when the waters! ach cop? (Toe spbericl eytenc of belghtsbown nig 11? bs olume, Ps nih 1S), bere 7 he ads of he spe] Bo y= 2H WI} =3aHP — (FN $0, ON 6a: DA— AH MA= ADAG I) Weare thd fet BVT, 40 L=MADAO-h), When Ro 1, Df Uisw ints, Az A meta bl of adie 90 senbnetes coated wth epson hick prof ie, whichis nling ache neo Cate cendoeter per hours Find there at whioy the tnchacs of the eet crating when the fe 10 etn e F Let be the cichases of toe ior. “The volams of the Kx ¥~ $ui0+ 8)'— Ha(0), So, DV He(O04KF Dh. We are told Wt B= o Se Heme, “2 (04 ADA, Whew = 10, ~2— (075, a> -uonmem 1449 A snowballs inereasingn wut atthe rite of Wan". How fasts the serfs area growing a he moment ‘Poon nection of the snowbal Sm? Ho tucnubice cen AaAert So, DA~Ser-Dp Now, Vo bur DW: fol ta DN 10, “Bo, MIke Dy Spear Dr =¥eDA. When banda, iar De We we Fees ma, 1444 then object t monng onthe sore! ya at wha pin lade y-cordnate of ube objess changing nee ‘nce me aptly han the >-oaais? HL. So,dhepoinsare(l, Qamd( 1, $9, (Other eral of time when the abject remais Fo pya3e-te Wea Ds=3-Pph, 221 felis ont when Dz=0," By 8” thi bappens within ned at ome point on te cave} MAS Ashe diagonal of w one acting a ate of 3 cubic aches per nuts, Bow fas is these ofthe cube ‘seating? F Leta be the Teagh of the dlagonal of a othe of se 2, They afr st Pang. Tus, 3-V3Ds,, Dam V3iaham 1446 The two equa ses uf wn louseles tangle wih xe Rowe ase deeming st he rate o( 3 inches per mine ow fat the aveu deccing when te two equal side ate oul to the bse? 7 Fig. eas