Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.8 - Exponential Growth and Decay - 3.8 Exercises - Page 243: 10

Answer

(a) The half-life is 12.3 years. (b) 28.5 years

Work Step by Step

(a) We can find the value of $k$: $m(t) = m(0)e^{kt}$ $m(1) = m(0)e^{k} = 0.945~m(0)$ $e^{k} = 0.945$ $k = ln(0.945)$ $k = -0.05657$ Then: $m(t) = m(0)~e^{-0.05657~t}$ We can find the time $t$ when only half remains: $m(t) = m(0)~e^{-0.05657~t} = 0.5~m(0)$ $e^{-0.05657~t} = 0.5$ $(-0.05657)~t = ln(0.5)$ $t = \frac{ln(0.5)}{-0.05657}$ $t = 12.3~years$ The half-life is 12.3 years. (b) We can find the time $t$ when only 20% remains: $m(t) = m(0)~e^{-0.05657~t} = 0.2~m(0)$ $e^{-0.05657~t} = 0.2$ $(-0.05657)~t = ln(0.2)$ $t = \frac{ln(0.2)}{-0.05657}$ $t = 28.5~years$
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