Susan finds an alien artifact in the desert, where there are temperature variati
ID: 3162091 • 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)=
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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)=
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(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)=
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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
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to
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degrees.
Explanation / 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) = 80,
A = 35 and P = 2pi, and so E(t) = 80 + 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 + (0.6)*T = (0.6)*(80 + 35*sin(t)) ------>
dT/dt + (0.6)*T = 48 + 21*sin(t),
which is a first-order linear ordinary differential equation.
It has the general solution
T(t) = A*e^[(-0.6)*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) = 80 then the solution turns out to be
T(t) = (15.4412)*e^[(0.6)*t] + (9.2647)*sin(t) - (15.4412)*cos(t) + 80.
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