# PP 7.5 - Exponential Growth and Decay Exponential Model If...

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Exponential Growth and Decay

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Exponential Model If y(t) is the value of a quantity at time t and if the rate of change of y with respect to t is proportional to its size y(t) at any time, then ky dt dy = k is a constant, called the constant of proportionality.
Comments If k is positive, then the model represents natural or exponential growth. If k is negative, then the model represents natural or exponential decay. ky dt dy = This is a differential equation because it involves y and its derivative.

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Theorem kt e y t y ) 0 ( ) ( = The only solutions of the exponential model are the exponential functions kt e y t y ky dt dy ) 0 ( ) ( : s other word In = =
Example 1. Determine an differential equation for Q(t). Exponential model: kQ dt dQ = The rate at which a radioactive substance decays is proportional to the amount Q(t) of the substance remaining at time t. 2. Find the solution of the equation, assuming that initially there were 15 kg of the substance and that its half-life is 120 years.

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Solution kt e Q t Q kQ dt dQ ) 0 ( ) ( = = 5 . 7 ) 120 ( and 15 ) ( = = Q
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