A thermometer is removed from a room where the temperature is 70 and is taken outside, where the air temperature is 10. After half a minute the thermometer reads 50. What is the reading of the thermometer at t=1 min? How long will it take for the thermometer to reach 15?
$\frac{dT}{dt}=k(T-T_m)$
$\frac{1}{T-T_m}dT=kdt$
$\int \frac{1}{T-T_m}dT=\int k dt$
$\ln{|T-T_m|}=kt+C_1$
$T=C_2e^{kt}+T_m$
Using the initial conditions when the thermostat and inside air are both 70 at t=0
$70=C_2 e^{k(0)}+70$ uh-oh
Where am I going wrong? The thermostat and initial ambience are both 70 aren't they?
By the way, I have a lot of trouble telling if k should be negative or positive (but it always becomes apparent in the end).
Best Answer
No the initial ambient is the outside temperature, you have: $$70=C_2e^{c\times 0}+10=C_2+10$$ so $C_2=60$.
Also the way you have set up your equations $k<0$ since you need $dT/dt<0$ when $T>T_m$ assuming $T_m$ is the ambient temperature.