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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/ ..~.
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(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
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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
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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 initialvalue 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 initialvalue problem from part (a). m is S9 m mt, w vs We)  Wm A = Ics
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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 wellstirred 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
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(c) At What time is the tank full?
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(d) How much salt is in the taﬂ'k‘rimgliﬂt'badre he tank starts to overﬂow? : Tit ‘ 7 L too) ’—
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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
initialvalue 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 initialvalue problem from part(a). Put your solution in explicit form. i_\
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 Math, Differential Equations, Equations

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