Heating a soup (solving this problem with a DE)

[JessicaHelena]

Homework Statement

A soup is heated from $0<t<t_1$ on an outdoor camp stove, and is cooled. The temperature $T(t)$ satisfies $T' + 0.1T = q(t)$ when $0<t<t_1$, and $T' + 0.1T = 0$ when $t1 < t$ where $q(t)$ represents the heat used to warm up the soup. How long will it take for the soup to be at $40$°C?

Relevant Equations

$\frac{Dt}{dt} = k(Te - T)$
$T(t) = Te + (T0 - Te)e^{-kt}$

I'm having quite a bit of a problem with this one. I've managed to figure out that $T_0 = 0$. However, not knowing what $q(t)$ is bothers me, although it seems that I could theoretically solve the problem without knowing it. For $t>t_1$, integration by parts gives me $T = Ce^{-t/10}$ where $C = T(t_1)$. And to get $T(t_1)$, I solve the inhomogeneous equation with $q(t)$, by letting $T = ue^{-t/10}$. THen I get that $u = \int q(t) e^{t/10} dt$. But where do I go from here?

 

[HallsofIvy]

You are already told that T'+ 0.1T= q(t) for t less than t1 and T'+ 0.1T= 0 for t larger than t1 so your "dT/dt= k(Te- T0)" is irrelevant. You are asked when the temperature of the soup will be 40 degrees.

You are correct that you can't solve this without knowing what the original temperature, at t= 0 or some other specific value of t. You say "I managed to figure out that T0 is 0". How? Was the initial temperature of the soup given? If the initial temperature was 0C then it was frozen. Was that given in the problem?

 

[JessicaHelena]

It wasn't given, but that was part of an earlier question which I frankly guessed and got right.

 

[JessicaHelena]

Might I be able to solve for q(t) from T0? I don't quite see a way for that, however...

 

[HallsofIvy]

No, q(t), the heat coming into the soup, is completely independent of any information you have given here and has to be given separately. You have already told us that your original post did not include all of the information you have. Perhaps q(t) was also "part of an earlier question".

 

[JessicaHelena]

@HallsofIvy
The earlier problem, as far as I can tell, is worded the exactly same way, but I will include it here:

 

[Dragon27]

You are asked when the temperature of the soup will be 40 degrees.

The question (if I googled the right problem) is a little bit different - one is asked to estimate the number of minutes for which the porridge will stay at the optimal temperature of 40°±1° (because Goldilocks won't eat the porridge that is too hot or too cold).
Edit: and there's also a hint that says to use the (differential) equation but not solve it (which is good enough for the estimate).