3.2.4 Determining the Proportionality Constant
         and Interpreting the Result

Recall that we don't actually know a value for `k` yet. Next we will see how our detective might determine a value of `k` that fits the problem being solved.

Activity 3

Suppose our detective makes a second measurement of body temperature at 9:50 A.M. and finds that the temperature has dropped to 28°C, that is, $T=28$ at $t=1$. How can we use that information to find `k`? What is `k`?

Comment on Activity 3

The numerical value comes from a pocket calculator, which any modern detective always has at hand. Figure 3 shows the slope field for $\frac{dT}{dt}=-k(T-21)$, now drawn with the known value of `k` along with the now fully-known solution through $(0,30)$.

We are now ready to use the solution of the initial value problem to estimate the time of death.

Activity 4

Use the solution function just obtained to determine the time of death. Once you know the time of death, make a conjecture about the likely murderer.

Comment on Activity 4