Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 8 - Cumulative Review Exercises - Page 952: 13


a) The decay model is $A\left( t \right)=900{{e}^{-0.017t}}$. b) The amount is $759.298\,\,\text{grams}$.

Work Step by Step

(a) Consider that at any time $t$, the amount of the radioactive substance present is $A\left( t \right)$, following the exponential decay model: $A\left( t \right)={{A}_{0}}{{e}^{kt}}$ Initially there are 900 grams of the substance present,$A\left( 0 \right)=900$, hence $\begin{align} & 900={{A}_{0}}{{e}^{0\times k}} \\ & {{A}_{0}}=900 \end{align}$ Therefore $A\left( t \right)=900{{e}^{kt}}$ The substance has a half-life of 40 days; hence for $t=40$, $A\left( 40 \right)=450$ Therefore: $\begin{align} & 450=900{{e}^{40k}} \\ & {{e}^{40k}}=\frac{1}{2} \\ & k=\frac{1}{40}{{\log }_{e}}\frac{1}{2} \\ & =-0.017 \end{align}$ Therefore, the decaying model for the substance is $A\left( t \right)=900{{e}^{-0.017t}}$ (b) The radioactive substance remains after 10 days, that is, for $t=10$ $\begin{align} & A\left( 10 \right)=900{{e}^{-0.017\times 10}} \\ & =759.298 \end{align}$ Thus, the radioactive substance that remains after 10 days is $759.298\,\,\text{grams}$.
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