College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.5 - Page 505: 31


about 5.5 hours

Work Step by Step

The decay model: $A=A_{0}e^{kt}$ ($ k < 0 )$ We find k from the half life information $0.5A_{0}=A_{0}e^{k\cdot 36}\qquad/\div A_{0}$ $0.5=e^{36k}\qquad $ ... apply ln( ) to both sides $\ln 0.5=36k$ $ k=\displaystyle \frac{\ln 0.5}{36}\approx$-0.0192540883489$\approx-0.0193$ The decay model is : $A=A_{0}e^{-0.0193t}$ We want t (in hours) when $A=0.90A_{0}$ $0.9A_{0}=A_{0}e^{-0.0193t}\qquad/\div A_{0}$ $0.9=e^{-0.0193t}\qquad $ ... apply ln( ) to both sides $\ln 0.9=-0.0193t\qquad \div(-0.0193)$ $ t=\displaystyle \frac{\ln 0.9}{-0.0193}\approx$5.459094075548$\approx 5.5$ hours
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