4.2.notes - A e k t T = 1 2 A e kt giving that e kT = 1 2...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng On exponential growth/decay models I plan to give a heuristic derivation of a formula for exponential decay in class. The formula is y ( t ) = A × ( 1 2 ) t/T , where A is the initial amount, y ( t ) is the amount remaining after t years and T is the half-life (in years). I wish to give a formal derivation here. Suppose that we have a radioactive element with half-life T years. The initial amount is A (in grams say) and the amount remaining t years later is modeled by y ( t ) = A e kt , (1) where A is the initial amount and k is a constant to be determined. Since at any time t , the amount is halved in T years, we have y ( t + T ) = 1 2 y ( t ) . Using equation (1), we see that y ( t + T ) = A e k ( t + T ) , and y ( t ) = A e kt . Since y ( t + T ) = 1 2 y ( t ), we have A e k ( t + T ) =
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Unformatted text preview: A e k ( t + T ) = 1 2 A e kt , giving that e kT = 1 2 , implying e k = ( 1 2 ) 1 /T . Thus, equation (1) becomes y ( t ) = A e kt = A (e k ) t = A × p ( 1 2 ) 1 /T P t = A × ( 1 2 ) t/T , which is precisely the formula we obtain in class. Similarly, if the value of a stock portfolio (or a home) is doubled every T years, then its future value y ( t ) in t years is related to its present value P by the formula: y ( t ) = P × 2 t/T . The formulas appear in Example 2 and Exercise 33 in § 4.1. If the base is b > 0 instead of 2 or 1/2, for example, y ( t ) = P × b t/ 10 , where t is in years for example , then the model says that (a) the initial vealue of y is P , and (b) the value of y is multipled by a factor of b every t = 10 years....
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