# How do we use exponential growth and decay in real life?

Apr 14, 2015

One of the "home grown" examples is a gradual diminishing of a temperature of a hot body down to room temperature. For instance, the kettle with boiling water cools down after the heater is turned off.

The physical law describing this process of cooling sounds like this:
The speed of a cooling of a hot object is proportional to a difference in temperature between the cooling body and environment, assuming that the environment is large enough to absorb the heat without really changing its own temperature.

In other words, the cooling is faster if the difference in temperatures between the object and the environment is greater and the speed of cooling diminishes to zero as the temperature of a hot object gradually diminishes to a temperature of an environment

Here is the reason why it leads to exponential decay.
Let the temperature of a hot body is a function of time $K \left(t\right)$, while the temperature of the environment is constant ${K}_{0}$.
The difference between the temperature of a body and an environment is $K \left(t\right) - {K}_{0}$.
The speed of cooling is, obviously, a derivative of a function $K \left(t\right)$ by time $t$, that is $K ' \left(t\right)$ or $\frac{\mathrm{dK} \left(t\right)}{\mathrm{dt}}$.

The physical law of changing the temperature of a cooling body mentioned above now looks like
$\frac{\mathrm{dK} \left(t\right)}{\mathrm{dt}} = - \alpha \cdot \left[K \left(t\right) - {K}_{0}\right]$
where $\alpha$ is some coefficient and a minus sign is used because we assume that the temperature $K \left(t\right)$ is higher than that of an environment ${K}_{0}$, but it's diminishing, therefore the derivative is negative, and we would like to have a constant $\alpha$ as some positive constant that describes physical properties of the environment (like how well it carries off the heat from a hot object).

Basically, what we have above is a differential equation (physics, as you know, is all about differential equations).

Its solution is
$K \left(t\right) = {e}^{- \alpha \cdot t} + {K}_{0}$

Indeed, the derivative of this function $K \left(t\right)$ is
$\frac{\mathrm{dK} \left(t\right)}{\mathrm{dt}} = - \alpha \cdot {e}^{- \alpha \cdot t}$
But ${e}^{- \alpha \cdot t} = K \left(t\right) - {K}_{0}$.
Therefore, $\frac{\mathrm{dK} \left(t\right)}{\mathrm{dt}} = - \alpha \cdot \left[K \left(t\right) - {K}_{0}\right]$ and our differential equation is satisfied.

So, the process of cooling of a kettle after the heat is off is a good example of an exponential decay.

This example prompts to a conclusion that every process with a speed of change proportional to its value exhibits the exponential dependency.
Another typical example is a population grows. Obviously, the speed of a population's grows (that is, the increment of the number of species), generally speaking, should be proportional to the number of species.