Alternative Formulas for Growth and Decay
Date: 08/02/2002 at 21:59:38 From: Jared Subject: Math Analysis/Algebra 2 I was just wondering about the Growth and Decay formula. Last year my teacher told me it was q = q_0 * e^(rt) My teacher this year says it's q = q_0 * a^t One of these says that the variable r decides the growth and decay and the other say that the variable a decides the growth or decay. And why does one equation include e and the other doesn't? I'm confused.
Date: 08/02/2002 at 23:41:51 From: Doctor Peterson Subject: Re: Math Analysis/Algebra 2 Hi, Jared. These two equations are equivalent; it's sort of like saying that you can measure your height in either feet or meters. Since we can write a as e^ln(a), the second equation is the same as q = q_0 e^(ln(a)t) and the r from the first equation is just ln(a). Which form you use depends on your point of view. The first form emphasizes the time constant r, which tells how long it takes to grow to e times the initial amount. This is the more sophisticated form, I suppose; the exponential with e makes calculus easier, such as finding the instantaneous rate of increase. The second form emphasizes the base a, which is how much you grow by in a unit of time. This links it to geometrical sequences. Another useful form is 2^(-t/h), where h is the half-life, the time it takes to reach 1/2 the original amount. Each of these has its own special use. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 08/04/2002 at 19:40:44 From: Jared Subject: Thank you (Math Analysis/Algebra 2) Thanks, I understand it now... you really helped!
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