6. Model radioactive decay using the notation t = time (independent variable), r(t) = amount of particular radioactive isotope present at time t (dependent variable), -λ = decay rate (parameter). Note that the minus sign is used so that > > 0. (a) Using this notation, write a model for the decay of a particular radioactive iso- tope. = (b) If the amount of the isotope present at t initial-value problem for the model in part (a). 0 is ro, state the corresponding
6. Model radioactive decay using the notation t = time (independent variable), r(t) = amount of particular radioactive isotope present at time t (dependent variable), -λ = decay rate (parameter). Note that the minus sign is used so that > > 0. (a) Using this notation, write a model for the decay of a particular radioactive iso- tope. = (b) If the amount of the isotope present at t initial-value problem for the model in part (a). 0 is ro, state the corresponding
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In Exercises 6–10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law:
The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used.
6. Model radioactive decay using the notation
\( t = \) time (independent variable),
\( r(t) = \) amount of particular radioactive isotope present at time \( t \) (dependent variable),
\(-\lambda = \) decay rate (parameter).
Note that the minus sign is used so that \( \lambda > 0 \).
(a) Using this notation, write a model for the decay of a particular radioactive isotope.
(b) If the amount of the isotope present at \( t = 0 \) is \( r_0 \), state the corresponding initial-value problem for the model in part (a).
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