According to Newton's law of cooling, the temperature $T$ of an object at time $t$ is given by $T = T_A + (T_0 - T_A)e^{-kt}$, where $T_A$ is the ambient temperature and $T_0$ is the initial temperature, and $k$ is the cooling rate constant. For a certain type of beverage container, the value of $k$ is known to be $0.025 \text{ min}^{-1}$
Assume that $T_{A} = 36^{\circ}$F exactly and that $T_0$ varies with mean $= 72^{\circ}$F and standard deviation $= 0.5^{\circ}$F. Find mean of $T$ and standard deviation of $T$ at time $t = 10$.
I know this is a physics problem but I got this problem from my statistics class and I have no idea how to approach this. What distribution model am I supposed to use here?
The expected value (i.e. mean) is a linear operator. This means we can write:
$$\langle T \rangle = \langle T_A + (T_0 - T_A)e^{-kt}\rangle = \langle T_A \rangle + \langle T_0 e^{-kt} \rangle - \langle T_Ae^{-kt} \rangle = T_A + \langle T_0 \rangle e^{-kt} - T_A e^{-kt}$$ (Try to think about why this would work from a conceptual standpoint. Adding in and multiplying by constants will just shift and re-scale the mean).
Standard deviation can be defined as: $$\sigma = \sqrt{\langle T^2 \rangle - \langle T \rangle ^2}$$
Do you see how to finish?