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Finding the Function that Describes Radioactive Decay · Physics Playground

Exercises Exponentials and logarithms

Example 2 from Algebra and Trigonometry, 6.7 Exponential and Logarithmic Models

Finding the Function that Describes Radioactive Decay

The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, $t .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
This formula is derived as follows.
$\begin{array}{l} A = A_{0} e^{k t} & The continuous growth formula . \\ 0.5 A_{0} = A_{0} e^{k \cdot 5730} & Substitute the half-life for t and 0.5 A_{0} for f (t) . \\ 0.5 = e^{5730 k} & Divide by A_{0} . \\ \ln (0.5) = 5730 k & Take the natural log of both sides . \\ k = \frac{\ln (0.5)}{5730} & Divide by the coefficient of k . \\ A = A_{0} e^{(\frac{\ln (0.5)}{5730}) t} & Substitute for r in the continuous growth formula . \end{array}$
The function that describes this continuous decay is $f (t) = A_{0} e^{(\frac{\ln (0.5)}{5730}) t} .$ We observe that the coefficient of $t,$ $\frac{\ln (0.5)}{5730} \approx - 1.2097 \times 10^{−4}$ is negative, as expected in the case of exponential decay.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The half-life of plutonium-244 is 80,000,000 years. Find a function that gives the amount of plutonium-244 remaining as a function of time, measured in years.

How did it go?

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From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 6.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.

Solution

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