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Unformatted text preview: Chapter 9 Exponential Growth and Decay: Differential Equations 9.1 A differential equation is an equation in which some function is related to its own derivative(s). For each of the following functions, calculate the appropriate derivative, and show that the function satisfies the indicated differential equation (a) f ( x ) = 2 e- 3 x , f prime ( x ) =- 3 f ( x ) (b) f ( t ) = Ce kt , f prime ( t ) = kf ( t ) (c) f ( t ) = 1- e- t , f prime ( t ) = 1- f ( t ) 9.2 Consider the function y = f ( t ) = Ce kt where C and k are constants. For what value(s) of these constants does this function satisfy the equation (a) dy dt =- 5 y , (b) dy dt = 3 y . [Remark: an equation which involves a function and its derivative is called a differential equa- tion.] 9.3 Find a function that satisfies each of the following differential equations . [Remark: all your answers will be exponential functions, but they may have different dependent and independent variables.] (a) dy dt =- y , v.2005.1 - September 4, 2009 1 Math 102 Problems Chapter 9 (b) dc dx =- . 1 c and c (0) = 20, (c) dz dt = 3 z and z (0) = 5. 9.4 If 70% of a radioactive substance remains after one year, find its half-life. 9.5 Carbon 14 Carbon 14 has a half-life of 5730 years. This means that after 5730 years, a sample of Carbon 14, which is a radioactive isotope of carbon will have lost one half of its original radioactivity. (a) Estimate how long it takes for the sample to fall to roughly 0.001 of its original level of radioactivity. (b) Each gram of 14 C has an activity given here in units of 12 decays per minute. After some time, the amount of radioactivity decreases. For example, a sample 5730 years old has only one half the original activity level, i.e. 6 decays per minute. If a 1 gm sample of material is found to have 45 decays per hour, approximately how old is it? (Note: 14 C is used in radiocarbon dating, a process by which the age of materials containing carbon can be estimated. W. Libby received the Nobel prize in chemistry in 1960 for developing this technique.) 9.6 Strontium-90 Strontium-90 is a radioactive isotope with a half-life of 29 years. If you begin with a sample of 800 units, how long will it take for the amount of radioactivity of the strontium sample to be reduced to (a) 400 units (b) 200 units (c) 1 unit 9.7 More radioactivity...
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at The University of British Columbia.
- Winter '09