I have recently seen a quote about determining how long a body has been dead:
“Dead bodies lose heat exponentially, and therefore e can be used in an appropriate equation to determine how long individuals have been dead” (Calvin Clawson, Mathematical Mysteries, 1996).
Does anyone have an idea about such equation?
On
This is an example of Newton's law of cooling, which says basically that in an environment, the temperatures of all bodies will average out. The law is given by a differential equation whose solution leads to an exponential function.
These two YouTube videos might help you out:
I don't know the specifics, but the equation will be something of the form $$ H=Ae^{-kt} $$ where $H$ is heat, $t$ is time since death and $A,k$ are (known) constants. Then, if you measure the heat $H$, you can find the time $t$ since death by: \begin{align} H&=Ae^{-kt}\\ \frac{H}{A}&=e^{-kt}\\ -kt&=\log\left(\frac{H}{A}\right)\\ t&=\frac{-1}{k}\log\left(\frac{H}{A}\right)\\ \end{align} I disgree that this method really involves the number $e$, though. Indeed, we could replace $e$ with any exponent $a$ and get the same thing, since: $$ a^{-kx}=\left(e^{\log(a)}\right)^{-kx}=e^{-k\log(a)x}=e^{-\hat{k}x} $$ where $\hat{k}=k\log(a)$. In other words, passing from $e$ to another exponent just changes the constant $k$.
$e$ is certainly the most natural exponent, though, for reasons that will become clear to you in the course of your mathematical education.