Math 2250-1 Newton's law of cooling application - Maple Project 1 Here's a nice picture of the various solutions to the IVP for our DE, with the Athens data. In fact, it's figure 1.5.10 in the text. I've included the Maple commands which created the display. Notice the syntax elements of these commands - how to define functions, that := is how to define something in Maple, the # lets us write comments without affecting the commands, etc. Notice too that here, Maple checks a lot of the work you'll be doing by hand in the assignment, using the "dsolve" command! (Although the answer is a mess it should agree with your hand work and the text after you do some algebra.) restart: #clear all old definitions from memory with(DEtools): #load the DE library of commands A:=t->a0 + a1*cos(omega*t) + b1*sin(omega*t); #formula for ambient temperature, with free #parameters a0, a1, b1, omega. This is equation #(1), but not using subscripts on the variables. #When you enter multi-line #commands hold down the shift key while you hit #"enter" or "return", to prevent premature execution

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deqtn3:=diff(u(t),t)=-k*(u(t)-A(t)); #I named the DE "deqtn3" since it's equation (3) #on top of page 57.

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dsolve({deqtn3,u(0)=0},u(t));

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a0:=80; #average ambient temp in Georgia in July a1:=-5; b1:=-5*sqrt(3.0); #the a1 and b1 values were worked out by hand, #using the cosine addition #angle formula, assuming 4 a.m. temp min and 4 p.m max, #and range from 70 to 90 degrees, #for trigonometric temp oscillation. omega:=Pi/12; #this makes the period equal to 24 (hours) k:=0.2; #constant for a well-insulated building

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DEplot(deqtn3,u(t),t=0..50,{[u(0)=65],[u(0)=70], [u(0)=75],[u(0)=80],[u(0)=85],[u(0)=90],[u(0)=95]}, arrows=line, color=black,linecolor=black, dirgrid=[30,30], stepsize=1, title="inside temperatures");

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We can also recreate a comparison of the steady-periodic indoor temperature vs. the outdoor temperatures, Figure 1.5.11 on page 58: with(plots): #load the plotting package usp:=t->80+(2.3351)*cos(Pi/12*t)-(5.6036)*sin(Pi/12*t): plot1:=plot(usp(t),t=0..50,color=black): plot2:=plot(A(t),t=0..50,color=black): display({plot1,plot2},title=`long-term indoor vs outdoor temperatures`);

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Notice that the indoor temperatures have a smaller amplitude, and that they lag the outdoor temperatures. The same thing happens with seasonal temperatures on earth, where the temperatures in a given hemisphere lag behind the times of maximal and minimal heating from the sun. Exercise 7: Find the phase delay (the difference, in hours), between the ambient temperature and the indoor steady periodic temperature. It is acceptable to approximate this from the display in Exercise 6 (clicking the mouse on a point of the display will yield the point coordinates in a small window at the top of your Maple program). It is even better if you can verify the phase delay by writing the steady-periodic solution in phase-amplitude form (see section 5.4 mechanical vibrations, page 324, for help). Exercise 8: Suppose the inside temperature is 76 degrees when the furnace is turned off. Using 3-d plots and other commands (see hints at Professor Gustafson's page, or use Maple help), figure out for the range of insulation constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRJDAuMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0Y1LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk0tRiw2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Yr LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkjbW9HRiQ2LVEiPEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFElMC40OEYnRjVGKw== , the times during the first 72 hours when the indoor temperature is at or below 30 degrees. (This could be bad for your water pipes.) Justify the logic and math used to find these "bad" times, in a short paragraph. Illustrate your work with a computer graphic.