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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 10,000 to 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

2 y 0 = y 0 e k t 2 = e k t ln 2 = k t t = ln 2 k .

Definition

If a quantity grows exponentially, the doubling time    is the amount of time it takes the quantity to double. It is given by

Doubling time = ln 2 k .

Using the doubling time

Assume a population of fish grows exponentially. A pond is stocked initially with 500 fish. After 6 months, there are 1000 fish in the pond. The owner will allow his friends and neighbors to fish on his pond after the fish population reaches 10,000 . When will the owner’s friends be allowed to fish?

We know it takes the population of fish 6 months to double in size. So, if t represents time in months, by the doubling-time formula, we have 6 = ( ln 2 ) / k . Then, k = ( ln 2 ) / 6 . Thus, the population is given by y = 500 e ( ( ln 2 ) / 6 ) t . To figure out when the population reaches 10,000 fish, we must solve the following equation:

10,000 = 500 e ( ln 2 / 6 ) t 20 = e ( ln 2 / 6 ) t ln 20 = ( ln 2 6 ) t t = 6 ( ln 20 ) ln 2 25.93.

The owner’s friends have to wait 25.93 months (a little more than 2 years) to fish in the pond.

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Suppose it takes 9 months for the fish population in [link] to reach 1000 fish. Under these circumstances, how long do the owner’s friends have to wait?

38.90 months

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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 k t .

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

y = k y 0 e k t = k y .

Rule: exponential decay model

Systems that exhibit exponential decay    behave according to the model

y = y 0 e k t ,

where y 0 represents the initial state of the system and k > 0 is a constant, called the decay constant .

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

This figure is a graph in the first quadrant. It is a decreasing exponential curve. It begins on the y-axis at 2000 and decreases towards the t-axis.
An example of exponential decay.

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

T = k ( T T a ) .

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 ( t ) = T ( t ) 0 = T ( t ) , and our equation becomes

y = k y .

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

Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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