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Solved: PROBLEM 8E(a) Consider the initial-value problem
Chapter 3, Problem 8E(choose chapter or problem)
(a) Consider the initial-value problem dA/dt = kA, \(A(0)=A_{0}\) as the model for the decay of a radioactive substance. Show that, in general, the half-life T of the substance is T = (ln 2)k.
(b) Show that the solution of the initial-value problem in part (a) can be written \(A(t)=A_{0} 2^{-t / T}\).
(c) If a radioactive substance has the half-life T given in part (a), how long will it take an initial amount \(A_{0}\) of the substance to decay to \(\frac{1}{8} A_{0}\) ?
Text Transcription:
A(0)=A_0
A(t) = A_0 2^-t/T
A_0
1/8 A_0
Questions & Answers
QUESTION:
(a) Consider the initial-value problem dA/dt = kA, \(A(0)=A_{0}\) as the model for the decay of a radioactive substance. Show that, in general, the half-life T of the substance is T = (ln 2)k.
(b) Show that the solution of the initial-value problem in part (a) can be written \(A(t)=A_{0} 2^{-t / T}\).
(c) If a radioactive substance has the half-life T given in part (a), how long will it take an initial amount \(A_{0}\) of the substance to decay to \(\frac{1}{8} A_{0}\) ?
Text Transcription:
A(0)=A_0
A(t) = A_0 2^-t/T
A_0
1/8 A_0
ANSWER:Step 1 of 5
In this problem we need to solve the given parts of the problem.