AK-MATH225F10-WS4

# AK-MATH225F10-WS4 - MATH225 Fall 2010 Name u 6me 3...

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Unformatted text preview: MATH225, Fall 2010 Name: ) u 6me 3 Worksheet 4 (2.6, 3.1, 3.2) Section: a O For full credit, you must ShOW all work and box answers. 1. Consider the initial value problem dy_ 2 _ =52: _: I/ ..~. "ﬁx/V X9 )4: n; qomhar a fin“: ﬁkrao;lq a [57am tit/‘1 (a) Use Euler’s Method, with a step size h 2 Ax = 0.25, to approximate the solution to the initial value problem on the interval 0 3 cc 3 1. You should not need any technology for this problem}. Lcyonai. ox calcu («fort problem. Adjust this spreadsheet, or create your own to = Act = 0.1, for 0 3X3 1. Print your results and attach approximate the initial value problem with step size h them to this worksheet. (c) Use your solutions from part (b) to answer y(0.1) z (pﬂOOO y(1)~ Ca, 305% TAB] EULER'S METHOD m2n=f(x_n,y_n) h m 10000000000 40000000000 0. _ 09000000000 07100000000 0. _ 0.1 0.2 o. o. 0.4 5 delta x I I |"" I! 1 — o. 0. mm o. 0. mm 0. mm 07754923929 02986115486 0. 08053535477 0.3514056632 0. Page PAGE] 2. A bacteria culture grows at a rate proportional to its size. (a) Write the differential equation that models this situation. Let P be the number of bacteria at time, t. (Hint: See Worksheet 2). P .— {61% ’1 P (b) Find an explicit solution to your differential equation from part(a) using either Separation of Variables or the Method of Integrating Factors. A P of :O'b, o +(_L)P;.o e ' 5F» Vkra '— ' ' L—ﬂ S P AP—SZAE / (fro 7‘s {Tonsil/Ti, labia!) I a”; {'K‘Jﬁ ’Lt Mr! 2 mm mm) 2%th :a C. C 115 p \$5750 + .z P wane/Ks mica/o W ~Lb ’ t: s... P :_ (0) Assume the population at time t = 0 is the constant Po. If it takes 2 hours for the popula lOIl 0 on e, ow long does it take for the population to triple? : {4n 2 Fro):4¢9=Po/HLPO P“) Pod P Lt) ' Po 6/ ‘5 ﬂ, = a ll; _ .. v Hal/75c L2}: QﬂB’LE’i .J. Can—:2 / Liz/Vol £3 7/83“: Acor‘sﬁt’SJW 5% 3. A cup of coffee is initially 165° F and is left in a room with an ambient te perature of 75° F. Suppose that after 10 minutes the coiiee is 160°F. Assume that Newton’s Law of Cooling applies: The rate of cooling is proportional to the difference between the current temperature and the ambient temperature. (a) Write an initial-value problem that models the temperature of the coffee. Let T represent the current temperature of the coffee at time, t, in minutes. iii: :KCT‘WQ / TCO):léS , TOO):/OO W (b) Find an explicit solution to the initial-value problem from part (a). m is S9 m- mt, w vs We) - Wm A = Ics \L‘Lﬂ‘v/ ’ ‘55 C T’:‘7\$ 7S (63512042 0 Ln ‘ T‘wg‘ :Kt+¢ \$0111 Kt a<int a a C b C /TCt> : +906 r rbs T375 = 3, 6L / /+ = lap A150 mo 7 Clog: \ ‘f' - (c) How long does it take for the coffee to reach 80°F? n L Kincaﬂn (/Zé _ iénéalg) “° — 80—75%ch TLt)=‘75+rc,Q Cyrus‘s) L t Z 4. At time t = 0 a tank contains 10 lb of salt dissolved in 100 gallons of water. Assume that water containing 1 lb of salt per gallon is entering the tank at a rate of 2 gallons per minute and the well-stirred mixture is draining from the tank at a rate of 1 gallon per minute. ‘T’kr. {M L as A Capach a‘l; ZOO’joi/lons . (a) Set up the initial value problem that models the dynamics of salt in the tank. ig§> t1 m; mo at £13?“ }V (0% ‘. 0x51500604 59‘“: 7’7 ﬁankwé) 3% , a 70’ lag, v : V0 1 omc vat) : 100 + 1562”) (b) Solve the initial value problem from part (a). 9K Scparaﬁioq of“ Varikbldﬁ I’m“; waré l .— _L it:- + (wow ‘ Z .J_._ At wort ML’é);C v0 #0071 U 00 +07 ; ‘Lliﬂom‘ﬂl-I-Q C. ———.—.. (7‘ : 'L‘lféloo‘t’t) +7 [Cort C .— Wo): 26 +7.2; “’0 r—ISOO WW Q00 +13); (c) At What time is the tank full? \/ 2 00 't i7 : —v Mtnr (d) How much salt is in the taﬂ'k‘rimg-liﬂt'badre he tank starts to overﬂow? : Tit ‘ 7 L too) ’— : - —~ 5 a 5. A 100—volt electromotive force is applied' to an E series circuit in W ich the resistance is 200 ohms and capacitance is 10“; farad. The following linear differential equation models the charge on the capacitor. dq Rd—t + ~ 1 '54 = EU) A goat?“ v50” Findethe charge q(t) on the capacitor if (1(0) : 0. 20014? 4- 10:3 6. (a) Suppose a student carrying a ﬂu virus comes to an isolated college campus of 5000 students. Determine an initial-value problem for the number of people a:(t) who have contracted the flu if the rate at which the disease spreads is proportional to the product of the number of students who have the flu and the number of students who have not been exposed to it. (Hint: The number of students who have not been exposed is 5000 + 1 —— :13. We are assuming everyone exposed contracts the ﬂu.) (Hint: See Worksheet 2.) time 01 ’ ____,_,__. x‘. no, at. 5604m55 was :— K¥(g0®+/-x‘)/ “0):, ham thc ﬂu 000 ~ ’ f Us! #3 Of’ ‘T WK ( . -—>< de (b) Solve the initial-value problem from part(a). Put your solution in explicit form. i_\ 65,, “r ((c'zn . 0': :xri‘otlds , g Ax : Kali Xaéo, stool [xzo/ X:5C3CD/ ’er ( Iﬂ’ﬂ’ﬂ F46??? tuiﬁsr‘ C emor‘éf, “DE 7'1; qmlﬂﬂrus .X \$052!"): ‘ I .1? ’ 500) _ _ _. Partinl Factions 3 5:0 J“ Sool”)() 4* ‘Zt ltd W Mean/whey : I Amen + goal/M ﬁbre ~ fear/115mm wig 4., / K /-.. v1,» V/‘Hbtol 'Scblﬁtr/ g<30! Q” 5W“ a C 5:4:531 ,4; 30%.“ b Egg/ﬂ I: goo/Lt + Soc/c. 6 C X EOOIC Soojw Eds/ex " 11¢ d x v 9630th 3} Exam goal—x ’ (- / ’4'[email protected] /o Some/t” Scotti: X 1 BOO/'4'; - X/k. ...
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