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117_pdfsam_math 54 differential equation solutions odd

117_pdfsam_math 54 differential equation solutions odd - m...

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Exercises 3.2 Thus D ( t )= t 30 +4 . The diameter D ( t ) of the snowball will be 2 inches when t 30 +4=2 t =60(m in)=1(h) , and the snowball will disappear when t 30 +4=0 t = 120 (min) = 2 (h) . 23. If m ( t )(w ith t measured in “days”) denotes the mass of a radioactive substance, the law of decay says that dm dt = km ( t ) , with the decay constant k depending on the substance. Solving this equation yields m ( t )= Ce kt . If the initial mass of the substance is m (0) = m 0
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Unformatted text preview: m ( t ) = m e kt . (3.2) In this problem, m = 50 g, and we know that m (3) = 10 g. These data yield 10 = m (3) = 50 · e k (3) ⇒ k = − ln 5 3 , and so the decay is governed by the equation m ( t ) = 50 e − (ln 5) t/ 3 = (50)5 − t/ 3 . After 4 days, the remaining amount will be m (4) = (50)5 − 4 / 3 g, which is (50)5 − 4 / 3 50 · 100% = 5 − 4 / 3 · 100% ≈ 11 . 7% of the original amount. 113...
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