math1a - quiz 6 - solns - Math 1A Quiz 6 Solutions and...

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Math 1A Quiz 6 Solutions and Grading March 14th, 2008 1. The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. a) Find the mass that remains after t years. b) After how long will only 1 mg remain? Solution to a): We use the equation y ( t ) = Ae kt and solve for the constants A and k . We’re given that y (0) = 100 and y (30) = 100 / 2 = 50. Hence: 100 = y (0) = Ae k · 0 = A 50 = y (30) = 100 e k · 30 1 / 2 = e 30 k ln(1 / 2) = 30 k ln(1 / 2) 30 = k So our final equation is y ( t ) = 100 e ln(1 / 2) 30 t = 100 e - ln(2) 30 t . Many people had just memorized the formula for the half-life equation, which is also perfectly fine, provided you memorize it correctly . Solution to b): Set 1 = y ( t ) = 100 e - ln(2) 30 t and solve for t : 1 / 100 = e - ln(2) 30 t ln(1 / 100) = - ln(2) 30 t - ln(100) = - ln(2) 30 t 30 ln(100) ln(2) = t. 1
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I saw this answer in many equivalent forms and they’re all fine. Grading:
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This note was uploaded on 09/17/2011 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

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math1a - quiz 6 - solns - Math 1A Quiz 6 Solutions and...

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