6.8 Exponential growth and decay  (Page 3/9)

If a quantity grows exponentially, the time it takes for the quantity to double remains constant. In other words, it takes the same amount of time for a population of bacteria to grow from $100$ to $200$ bacteria as it does to grow from $\mathrm{10,000}$ to $\mathrm{20,000}$ bacteria. This time is called the doubling time. To calculate the doubling time, we want to know when the quantity reaches twice its original size. So we have

Exponential decay model

Exponential functions can also be used to model populations that shrink (from disease, for example), or chemical compounds that break down over time. We say that such systems exhibit exponential decay, rather than exponential growth. The model is nearly the same, except there is a negative sign in the exponent. Thus, for some positive constant $k,$ we have $y=y_0{e}^{\text{−}kt}.$

As with exponential growth, there is a differential equation associated with exponential decay. We have

The following figure shows a graph of a representative exponential decay function.

Let’s look at a physical application of exponential decay. Newton’s law of cooling says that an object cools at a rate proportional to the difference between the temperature of the object and the temperature of the surroundings. In other words, if $T$ represents the temperature of the object and $T_a$ represents the ambient temperature in a room, then

Note that this is not quite the right model for exponential decay. We want the derivative to be proportional to the function, and this expression has the additional $T_a$ term. Fortunately, we can make a change of variables that resolves this issue. Let $y(t)=T(t)-T_a.$ Then $y^{\prime }(t)=T^{\prime }(t)-0=T^{\prime }(t),$ and our equation becomes

From our previous work, we know this relationship between y and its derivative leads to exponential decay. Thus,