The cooling curve of a cup of water

A cup of hot water was left to cool in a room at $24\ ^\circ\text{C}$, and its temperature was read every two minutes. You will make the program test Newton's law of cooling: if it holds, the logarithm of the excess temperature falls in a straight line, and its slope gives the cooling constant.

The idea. $T - T_0 = (T_s - T_0)e^{-kt}$, so $\ln(T - T_0) = \ln(T_s - T_0) - kt$: a straight line of slope $-k$.

Your tasks.

1. Work out the excess temperature excess at every reading.
2. Fit a straight line to np.log(excess) against t, and find k.
3. Use your k to predict when the water reaches $35\ ^\circ\text{C}$.