According to Newton’s law of cooling, the rate at which an object cools (or warms) is directly proportional to the temperature difference between the environment and the object itself. Three years ago the corpse of Dr. Deadman was discovered in the coroner’s office. The room temperature of the coroner’s office was ($17^\circ \text{C}$). The doctor’s body temperature was measured to be $27^\circ \text{C}$, which was $10^\circ \text{C}$ below normal.

1. Dr. Deadman’s body was found at 5:05 p.m. An hour later the body had cooled to $26^\circ \text{C}$. At what rate was the body cooling when it was found?

   Step 1 (a):

   $\frac{dT}{dt}=-k(T-17)$

   Step 2 (a):

   Solve the differential equation: $T(t)=Ce^{-kt}+17$
   Now use $(0,27)$ and $(1, 26)$ to solve for the values of the parameters.

   More Help (a):

   What is $T^\prime (0)$?

2. Approximately when was Dr. Deadman killed?

   Hint (b):

   Normal body temperature is $37^\circ \text{C}$.