Putting something with a different temperature in a space with a constant temperature. The assumption underlying here is that the overall room temperature stays constant (i.e. the thing that’s cooling is so small that it doesn’t hurt room temperature).

\begin{equation} y’(t) = -k(y-T_0) \end{equation}

where, \(T_0\) is the initial temperature.

The intuition of this modeling is that there is some \(T_0\), which as the temperature \(y\) of your object gets closer to t. The result we obtain

## Solving

\begin{equation} \int \frac{\dd{y}}{y-T_0} = \int -k \dd{t} \end{equation}

we can solve this:

\begin{equation} \ln |y-T_0| = -kt+C \end{equation}

which means we end up with:

\begin{equation} |y-T_0| = e^{-kt+C} = e^{C}e^{-kt} \end{equation}

So therefore:

\begin{equation} y(t) = T_0 + C_1e^{-kt} \end{equation}

to include both \(\pm\) cases.

this tells us that cooling and heating is exponential. We will fit our initial conditions rom data to obtain \(C_1\).