#$&*
Mth 279
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Practice test question
** **
Sorry, I forgot to include my access code and other information.
This is a problem from the practice test and I can't quite figure it out.
2. The temperature of a room approaches the outdoor temperature at a rate proportional to the
difference between the two temperatures. The outdoor temperature is -10 Celsius, and the initial
room temperature is 20 Celsius. The average rate at which temperature changes during the first
30 minutes is -.2 Celsius / minute.
Write the differential equation for this situation, and use it along with the given conditions to
find the temperature of the room as a function of clock time.
****
T_o = -10, T_r = 20, dT/dt = -.2
I think this is the equation:
dT_r/dt = T_r + (T_r - T_o)e^(kt)
-.2 = 20 + (30) e^(30k)
-20.2 = 30e^(30k)
ln(-20.2/30) = 30k
k = 1/30 ln (-20.2/30)
This seems horribly complex. And wrong.
dT_r/dt = T_r + (T_r - T_o) e^[ 1/30 ln (-20.2/30) ]
#####
I had the equation completely wrong
theta' = k(S - theta)
theta' + k*theta = Sk
e^(kt) theta' + ke^(kt) theta = Sk e^(kt)
integrate
e^(kt) theta = Se^(kt) + C
theta = S + Ce^(-kt)
plugging in initial values
20 = -10 + C
C =30
theta = -10 + 30e^(-kt)
However, I don't know where to go from here. Where would I plug in dtheta/dt = -.2 ???
I can't plug it into theta' = k(S-theta) because I don't know k
#####
** **
Is this separable???
** **
self-critique #$&*
#$&* self-critique
self-critique rating
rating #$&*:
@&
y is proportional to x if there exists a constant number k such that y = k x.
The rate of change of the temperature is dT / dt. The stated difference in temperatures is T - (-10 CelsiusO = T + 10 Celsius.
To say that this rate is proportional to the stated difference is to say that
dT/dt = k * (T + 10 Celsius).
This is the differential equation you need to solve, and it is separable. I don't think you'll have any trouble solving it, but of course you want to be sure you know how to write the equation from the given problem statement.
*@
@&
y is proportional to x if there exists a constant number k such that y = k x.
The rate of change of the temperature is dT / dt. The stated difference in temperatures is T - (-10 CelsiusO = T + 10 Celsius.
To say that this rate is proportional to the stated difference is to say that
dT/dt = k * (T + 10 Celsius).
This is the differential equation you need to solve, and it is separable. I don't think you'll have any trouble solving it, but of course you want to be sure you know how to write the equation from the given problem statement.
*@