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1 Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer (a a consan emperaure of 1 ), and he oher is o microwave hem a a LOW power. By coincidence, i urns ou ha he heaing ime is he same for boh mehods eacly 2 minues. Of course we can use eiher one bu suppose we can also use a combinaion of he wo one mehod for a cerain ime and hen he oher. Here s he problem can I lower my cooking ime by swiching? And if so how much? Of course we assume he swich is insananeous wih no loss of emperaure. The graph a he righ provides he T- (emperaure-ime) rajecories for boh mehods over he inerval 2. The difference beween he graphs come from he differen ways in which hea is ransferred o he egg. In he microwave oven, hea is absorbed by he egg a a consan rae, and as a consequence, is emperaure increases a a consan rae. However, in he boiling waer, he egg absorbs hea more quickly a he beginning when i is cool han laer when i is close o he emperaure of he waer. (a) (graphical) Find a consrucion on a copy of he graph which provides a soluion o he problem of how o shif beween he wo mehods in order o minimize he oal cooking ime. emperaure T (degs) cooking rajecory boiling waer B () microwave M ( ) ime (mins) Now here are differen ypes of argumen for an opimizaion problem like his. Some are local and aemp o argue ha your find your minimum simply by arranging o do he bes you can a every momen. Ohers are global and work by consrucing and comparing enire rajecories. Regardless of he ype of argumen you produce, make sure ha a he end you draw on he original se of aes he emperaure-ime rajecory of he egg if i is heaed in he opimal (shores ime) manner. Show clearly how your graph gives us he resuling minimum cooking ime. (b) (algebraic and numerical). The mahemaical form of he boiling waer curve follows from Newon s Law which saes ha he difference D in emperaure beween he waer and he egg is an equaion of eponenial decay. Use his o find an equaion for he boiling waer graph B(). Use his equaion o obain a rough numerical check of your resuls obained graphically, in (a). (c) (calculus). Use he ools of calculus o find he eac opimal swich poin and hereby check your answers o (a) and (b). 4.1 alligaor egg page 1

2 Soluion. This is a good problem because here are differen ways o make he argumen, graphical, numerical and analyic, and boh local and global. Wih luck, some sudens will use one approach and some he oher, and when hese are pu ogeher a he end, some imporan undersanding will be gained. (a) The local argumen. In is simples form, he local argumen assers ha we always wan he rae of increase of emperaure o be as big as possible, and ha s given by he slope, so we always wan o be on he graph ha has he bigges slope. Now if you look a he wo graphs, he microwave graph always has slope 4, bu he boiling-waer graph sars wih a high slope, greaer han 4, and ends wih a low slope, less han 4. So ha ells us o sar wih he high slope of he boiling waer graph, and sick wih i unil he slope has dropped o 4, and a ha poin swich o he microwave So he swich poin will be he poin on he boiling-waer graph wih slope 4. Tha means ha he angen a ha poin is parallel o he microwave graph. And ha seems o happen a around =8. Maybe a iny bi above. Say 8.2. Tha s when we ransfer i from he waer o he microwave. So wha is he resuling minimum cooking ime? Le s work i ou. We sar wih 8.2 minues in he waer, and ha brings i o a emperaure of abou 64 (read from he graph), so i has 26 o go o ge o 9, and a 4 /min ha ll ake 6.5 minues. So he oal is = We ge an overall cooking ime of abou 14.7 minues. Tha s a saving of more han 5 minues from he wo pure sraegies Wha made his eceedingly simple argumen work was he fac ha we could swich ovens a any ime a no cos in erms of eiher los ime or emperaure loss. So here s no reason no o epec o be always in he mode ha has he highes rae of emperaure increase for he curren emperaure. A global argumen. A few sudens seem more naurally o go righ away for a global argumen. For his we need o find a mehod ha acually looks a and compares he enire emperaure-ime rajecory for differen sraegies. 4.1 alligaor egg page 2

3 Here s an eample of such an argumen. Suppose ha we sar in he boiling waer, keep he egg here for minues, and hen swich he egg o he microwave and keep i here unil i s done. Wha does he resuling emperaure-ime rajecory look like? Tha is, draw he graph of he acual emperaure of he egg agains ime. Okay. The graph will depend on he value of he swich poin. To have a paricular eample, ake = 4. Tha means we follow he waer graph ill ime = 4. Bu hen wha? Well hen we swich o he microwave where he emperaure increases a a consan rae of 4º per minue. Thus, from =4 on, he graph is a sraigh line of slope 4. This las piece of he graph is a line parallel o he microwave graph, aking off from he =4 poin on he waer graph. Where does his line end? The egg is cooked when i aains 9º and hus he end poin of his line is found where i has heigh 9. This seems o be close o ime = 16. Thus he oal cooking ime is 16 minues. Tha was a paricular eample, bu i illusraes he general form. For any swich poin, he las par of he graph is always a line of slope 4 aking off from he swich poin. And he graph ends when he line ges o heigh 9. Now le s consider he opimizaion problem. We wan he shores cooking ime, and ha means we wan ha final piece of sraigh line o aain heigh 9 as far o he lef as possible. I should be clear ha his will happen when he line is as high as possible so wha we wan is he highes possible line of slope 4 which inersecs he waer graph. One way o find ha is o ake he microwave line and keep raising i unil i no longer inersecs he waer graph. I should be clear ha a his final poin i will be angen o he waer graph. Thus he opimal swich poin is he poin where he graph has slope 4. This gives us he same poin 8.2 ha we found wih he local analysis. Our opimal rajecory follows he boilingwaer curve o he angen poin and hen moves up along he angen. Wha is he oal cooking ime for his swich poin? i s where he line his T = 9 and ha s seen o be beween = 14 and = 15, somewha closer o 15. Tha s also in line wih our esimae of 14.7 from he local argumen. I s imporan o noice ha he same essenial consrucion appears in boh he local and he global argumens. Bu he sory is quie differen in he wo cases alligaor egg page 3

4 (b) We are given ha he difference D in emperaure beween he boiling waer (1º) and he egg B() is an equaion of eponenial decay of he form D() = 1 B() = Ar where r < 1. To evaluae he consans, use he wo endpoins of he graph, = and =2: D() = 1 B() = 1 1 = 9 D(2) = 1 B(2) = 1 9 = 1 This gives us he equaions: Ar = 9 Ar 2 = 1 Solving: A = 9 9r 2 = 1 r 2 = 1/ D () B () / 2 r = (1/ 9) =.896 Hence: D() = Ar = 9(.896) and he boiling waer emperaure equaion is: B() = 1 D() = 1 9(.896). An eponenial equaion always has a consan muliplier. Here he muliplier is.896. The emperaure difference D is cu by 1.4% every minue. Sudens end o prefer o work wih decimals, as above, bu my own preference is ofen for a more eac form and ha s he case here (parly because he numbers work ou). I would wrie: = 1/ 2 r = (1/ 9) ( 1/ 2 / 2 (1/ 9) ) (1/ 9) r = 1 B( ) = 1 Ar = In fac, I can resis a simplificaion: 1 B( ) = /1 / 2 This displays, no he 1-minue muliplier, bu he 1-minue muliplier. Over any 1 minue period, he emperaure difference beween he waer and he egg is muliplied by 1/ alligaor egg page 4

5 Finally, since we have an equaion for he B-graph, we can use his o check our graphical answer for he swich poin. This is he poin a which he slope of he B-graph is he same as he microwave slope: 4 /minue. Calculus will give us he ools of he derivaive o do his (par (c)) bu even wihou calculus we can ge a good idea from he slopes of shor secans (which are average raes of change). In par (a) we esimaed he slope-4 poin o be close o = 8.2. A he righ, we abulae he values of B() beween =8 and =9. On an inerval of lengh.1, we wan o look for an increase of.4. We ge eacly ha (o 3 decimal places) on he inerval [8.2, 8.3]. Tha ells us ha he swich poin will be somewhere in his inerval. [Acually a small argumen is needed for his. Can you find i?] We see ha on he previous inerval [8.1, 8.2] he change is a bi more han.4, and on he following inerval [8.3, 8.4] he change is a bi less han.4. Tha fis wih he concave-down form of he graph. (c) Our problem is o find he swich poin which minimizes he oal cooking ime. Le s give his a name: we ll call i Toal cooking ime: τ = τ(). Now we need o find an epression for τ(). Well i s he sum of he imes he egg spends in each phase. The boiling waer is easy enough minues. The microwave requires a bi more hough. The firs phase brough he emperaure of he egg up o B(). To finish cooking, we need an era 9 B() degrees, and a 4 /minue his will ake (9 B())/4 minues. The oal ime is hen 9 B( ) τ ( ) = + 4 The calculus approach is o say ha a an inerior minimum he derivaive of τ mus vanish: B ( ) τ ( ) = 1 = 4 This solves as B ( ) = (.896) = 4 (.896) =.45 To solve his for we ake he naural log of boh sides: ln(.896) = ln(.45) = ln.45 ln.896 = We should swich o he microwave afer 8.24 minues. This confirms our graphical analysis ime emperaure B() The derivaive condiion will always find an inerior minimum, bu his mus be checked agains he end-poins, = and =2, bu we already know ha hese can be opimal as hey boh give cooking imes of 2 minues (see Problem 1). The derivaive of B From our calculaions above: B ( ) = 1 9(.896) B ( ) = 9(.896) ln(.896) = 9.88(.896) 4.1 alligaor egg page 5

6 Problems 1. Use he fac ha B(1) = 7 o argue immediaely ha he end-poin soluions, = and =2, can be opimal. 2. Recen advances in cooking have shown ha alligaor eggs can safely be heaed in a microwave a a slighly higher seing, giving a emperaure increase of 5 /min, hus requiring only a 16-minue cooking ime. Supposing ha he boiling waer mehod is sill available, follow he various approaches of he eample o find he opimal poin o swich from he boiling waer o he microwave. Illusrae your seps on a copy of he graph, as is done in he Eample, and esimae he oal cooking ime. Full eplanaions should be given. 3. I has been deermined ha he delicae flavour of an alligaor egg will be bruised if he rae of change of emperaure of he egg ever eceeds 8 /minue. Use a graphical approach o find he cooking program which minimizes he oal ime, bu does no allow he rae of emperaure increase o eceed 8 /minue. As usual you can assume ha swiches beween mehods happen insananeously wih no emperaure loss. 4. I have a porable heaing coil which I can plug in (say in a moel room) and pu ino a large mug of waer and hea i up o make ea. If I pop a super ea cozy over he mug while I m doing his, hen here is no hea loss o he room, and he waer emperaure increases a a consan rae, going from 2 o 1 in 5 minues. (a) Draw he graph of he waer emperaure agains ime. (b) Now suppose ha I ve go he waer o 1, bu insead of making he ea, I ge waylaid by a desire o perform an eperimen and I simply leave he mug sanding in he room wihou he cozy. Draw wha you hink is a reasonable graph of he emperaure of he waer over a 4-minue period. Here are a couple of poins you can mark on your graph during he firs wo minues, he emperaure drops from 1 o 7, and during he second wo minues i drops o. (c) Now, wih he waer emperaure a, I pu he heaing coil back in he mug and aemp o hea i back up o 1 bu wihou he cozy. Do I succeed? No! I find ha afer a long ime he emperaure of he waer sabilizes a a level somewha shor of boiling. Show ha an analysis of your wo graphs will allow you o predic he emperaure a which my mug of waer will sabilize. 5. I have a hole in my ire which makes i lose pressure a he consan percenage rae of 4% every half minue. Saring a 4 kpa, he graph a he righ shows he pressure rajecory over a 2 minue period. Suppose ha, in spie of he hole, I ry o keep he ire inflaed by pumping air in a a slow consan rae. Wha I know is ha wihou he hole, he pump will inflae he ire from o 4 kpa in 2 minues. Bu if I use he same pump on he ire wih he hole, saring a kpa, i never reaches 4 kpa bu approaches a consan pressure. Draw a rough graph of he pressure agains ime for his case, eplaining how you have deermined he limiing pressure. Then use he ools of calculus o calculae his limiing pressure eacly. pressure P (kpa) ime (min) 4.1 alligaor egg page 6

7 Working copy of alligaor egg graph alligaor egg page 7

8 Working copy of ire pressure graph (problem 5). pressure P (kpa) ime (min) 4.1 alligaor egg page 8

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