Example of Radioactive Decay 1

Calculus/ODEs: Carbon-14 has a half-life of 5,715 years. A fossil has lost 75% of its original amount of C-14. How old is the fossil? How much C-14 remains after 50,000 years?

[MathDoctorBob]

@JarrettRandall Understood. The manipulation is Algebra, but the subject is definitely Calculus/ODEs. Students tend to have their hands full with manipulating the equation at first.

Here's the calculus:
Working backwards, if y = Ae^{rt}, then y' = Are^{rt} by the chain rule for e^x. So y'-ry = 0.

Starting with y'-ry = 0, we have y' = ry or y'/y = r. Integrating both sides,
ln(y) = rt+C => y = e^ln(y) = e^{rt+C} => y = e^C e^rt = Ae^{rt}.
-Bob

[DannyHart]

Thank you very much, this helped me a lot :)

[MathDoctorBob]

This appears to have been controversial in the 50s and 60s. Other scientists noted calibration issues by comparing with sources other than Libby's. From site c14dating

"Later measurements of the Libby half-life indicated the figure was ca. 3% too low and a more accurate half-life was 5730±40 years. This is known as the Cambridge half-life. (To convert a "Libby" age to an age using the Cambridge half-life, one must multiply by 1.03)."

[MathDoctorBob]

@DannyHart4082 You're welcome. Thanks for the comment.

[MathDoctorBob]

@Waymanator123 Yes, only no-budget solutions here. - Bob

[MathDoctorBob]

@tubbythegreat1 You're welcome! - Bob

[Waymanator]

I assume you used IMovie? (Mac program)

[ratchetx]

why can c14 only determine the age of objects less than 60000 yrs, nice vid pls help