Chapter 6 | Applications of Integration
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Rule: Exponential Decay Model Systems that exhibit exponential decay behave according to the model y = y 0 e − kt , 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.
Figure 6.80 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 ′ =− ky . From our previous work, we know this relationship between y and its derivative leads to exponential decay. Thus, y = y 0 e − kt , and we see that T − T a = ⎛ ⎝ T 0 − T a ⎞ ⎠ e − kt T = ⎛ ⎝ T 0 − T a ⎞ ⎠ e − kt + T a where T 0 represents the initial temperature. Let’s apply this formula in the following example. Example 6.45 Newton’s Law of Cooling
According to experienced baristas, the optimal temperature to serve coffee is between 155°F and 175°F. Suppose coffee is poured at a temperature of 200°F, and after 2 minutes in a 70°F room it has cooled to
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