An initially 100-gram sample of a radioactive substance decays according to the continuous exponential model \(N(t) = 100e^{-kt}\), where \(N(t)\) is the remaining mass in grams after \(t\) years and \(k\) is the decay constant. Given that the substance has a half-life of 1500 years, answer the following:
(a) Determine the decay constant \(k\).
(b) Calculate the remaining mass after 5000 years.
(c) Find the time required for the sample to decay to 20 grams.
Question
Determine the decay constant \(k\) given the half-life of 1500 years.
Hint
Hint
Hint
Set up the equation \(100e^{-1500k}=50\) and solve for \(k\).
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Complete Answer
\(k = \frac{\ln 2}{1500}\) (approximately 0.0004621 per year)
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Question
Calculate the remaining mass after 5000 years using the decay constant found in part (a).
Hint
Hint
Hint
Substitute \(t = 5000\) and \(k = \frac{\ln 2}{1500}\) into \(N(t)=100e^{-kt}\).
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- Formula Booklet
Complete Answer
\(N(5000) \approx 9.9\text{ grams}\)
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Question
Find the time required for the sample to decay to 20 grams.
Hint
Hint
Hint
Set \(100e^{-kt} = 20\) and solve for \(t\) using \(k = \frac{\ln 2}{1500}\).
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- Formula Booklet
Complete Answer
Approximately \(3483\) years
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