Susan finds an alien artifact in the desert, where there are temperature variati
ID: 2880618 • Letter: S
Question
Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in the 100s in the day. She is interested in how the artifact will respond to faster variations in temperature, so she kidnaps the artifact, takes it back to her lab (hotly pursued by the military police who patrol Area 51), and sticks it in an "oven" -- that is, a closed box whose temperature she can control precisely.
Let T(t) be the temperature of the artifact. Newton's law of cooling says that T(t) changes at a rate proportional to the difference between the temperature of the environment and the temperature of the artifact. This says that there is a constant k, not dependent on time, such that T=k(ET), where E is the temperature of the environment (the oven).
Before collecting the artifact from the desert, Susan measured its temperature at a couple of times, and she has determined that for the alien artifact, k=0.75.
Susan preheats her oven to 75 degrees Fahrenheit (she has stubbornly refused to join the metric world). At time t=0 the oven is at exactly 75 degrees and is heating up, and the oven runs through a temperature cycle every 2 minutes, in which its temperature varies by 35 degrees above and 35 degrees below 75 degrees.
Let E(t) be the temperature of the oven after t minutes.
E(t)=
equation editor
Equation Editor
At time t=0, when the artifact is at a temperature of 45 degrees, she puts it in the oven. Let T(t) be the temperature of the artifact at time t. Then T(0)=
equation editor
Equation Editor
(degrees)
Write a differential equation which models the temperature of the artifact.
T=f(t,T)=.75(75+35sin(t)-T) .
Note: Use T rather than T(t) since the latter confuses the computer. Don't enter units for this equation.
Solve the differential equation. To do this, you may find it helpful to know that if a is a constant, then
sin(t)eatdt=1a2+1eat(asin(t)cos(t))+C.
T(t)=
equation editor
Equation Editor
After Susan puts in the artifact in the oven, the military police break in and take her away. Think about what happens to her artifact as t and fill in the following sentence:
For large values of t, even though the oven temperature varies between 40 and 110 degrees, the artifact varies from
equation editor
Equation Editor
to
equation editor
Equation Editor
degrees.
the questons remain unanswured point) Susan finds an al on artract n the desert, where thore are temperature vanations from a low in the 30s at nght to a bagh in the 100s n the day she is interested in how the tect wil espond to faster vanations in temperatuur so she kdaps the artMact, takes t back to her lab ooty pursued by the mltary police who patrol Aea 51. and stocks in an ven" that is, a closed box whose temperature she can control precisely t T() be the temperature the atiact Nevron's law ot cooling says that T() changes at a rate proporoonal to the drererca between the temperature o the erwronment and the mperature o the artifact Ths say that thera aconstart k, not drpendert on time, such that T (E-T. where Eis the temperature the (he even) tore colecting the atract som the desert, susan measurod rstemperature at a coupla times, and she has determned that for the alen antact k 0.75 usan Preheats her Dven to 7s degrees Fahrenhet (she has stubbornly reused to yon the metnc world) tmet 01he oven s at eacty Ts degrees and s heatng up and the ven runs through a temperature cycle every 2m mnutes, in which its temperature wries by 35 degrees above and 35 degrees below 75 degrees. et E() be the temperature of the oven atert mvnutos he oven Let T() be the temperature of the attact at time t Then T0) emperature of 45 degrees, she puts when the artifact is al a ttt (degree) Write a dferential equal on which models the temperature of the attact Nolo Use Trather than TO since the atter corfuses the computer Don't enter unts for this equation Save the diferential equation To do this, you may fnd it helpful to know that fa is a constart, then e (asin() cos(t) +C. sin (t) dt 1 T(t) After Susan puts in the atract in the oven, the military polce break in and take her away Think about what happens to her arttact as t-+ oo and st n the folowng sent anca For largo values of t evan though the oven temperature vanas botwven 40 and 110 dogreos. the atfact vanas from degrees Note: You can earn parful oredit on this problemExplanation / Answer
The general formula is E(t) = E(0) + A*sin((2pi/P)*t) where
A is the amplitude and P the period. We are given that E(0) = 75,
A = 35 and P = 2pi, and so E(t) = 75 + 35*sin(t).
Do you also want to find the temperature of the artifact at time t?
If so, then the differential equation is now
dT/dt = k*(E - T) =>
dT/dt + k*T = k*E =>
dT/dt + (0.75)*T = (0.75)*(75 + 35*sin(t))
=> dT/dt + (0.75)*T = 56.25 + 26.25*sin(t),
which is a first-order linear ordinary differential equation.
It has the general solution
T(t) = A*e^[(-0.75)*t] + B*sin(t) + C*cos(t) + D,
where A,B,C and D are constants that can be determined
once we are given some initial conditions.
If we assume that T(0) = 75 then the solution turns out to be :
T(t) = (16.8)*e^[(-0.75)*t] + (12.6)*sin(t) - (16.8)*cos(t) + 75 [ Answer ]
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.