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section 6.4: using logarithms to solve exponential equations & section 6.5: exponential models
score: 35/53 answered: 9/13
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question 10
an initial sample of 405 g of a radioactive substance decays according to the function $a(t) = 405e^{-0.015t}$ where $t$ is given in years.
(a) how many grams of the substance will remain after 98 years? round your answer to one decimal place.
93.1 grams
(b) what is the half-life of the substance? round your answer to one decimal place.
years
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Explanation:

Step1: Calculate remaining mass at t=98

Substitute t=98 into \(A(t)=405e^{-0.015t}\):
\(A(98)=405e^{-0.015×98}=405e^{-1.47}\)
\(e^{-1.47}≈0.230\), so \(405×0.230≈93.15≈93.1\)

Step2: Find half-life (A=202.5)

Set \(202.5=405e^{-0.015t}\), divide by 405:
\(0.5=e^{-0.015t}\), take natural log:
\(\ln(0.5)=-0.015t\), solve for t:
\(t=\frac{-\ln(0.5)}{0.015}=\frac{\ln2}{0.015}≈\frac{0.6931}{0.015}≈46.2\)

Answer:

(a) 93.1
(b) 46.2