ACT (Enhanced)MathematicsMedium
A paleontologist is dating a fossil using carbon-14. The amount of carbon-14 remaining in a sample, A(t), after t years, is given by the formula A(t) = A0 * (0.5)^(t/5730), where A0 is the initial amount. If a fossil contains 12.5% of its original carbon-14, approximately how old is the fossil?
- A17,190 years
- B11,460 years
- C22,920 years
- D5,730 years
Show answer & explanationAnswer & explanation
Correct answer: A. 17,190 years
If 12.5% remains, then A(t)/A0 = 0.125. So, 0.125 = (0.5)^(t/5730). We know that 0.125 = 1/8 = (0.5)^3. Therefore, (0.5)^3 = (0.5)^(t/5730). This implies 3 = t/5730. Solving for t, we get t = 3 * 5730 = 17,190 years.
Why the other options are wrong
- B. This is two half-lives (25% remaining).
- C. This is four half-lives (6.25% remaining).
- D. This is one half-life (50% remaining).
Half-Life Calculation
Half-life is the time required for a quantity to reduce to half its initial value. It's often used in exponential decay models.
- Amount remaining = Initial Amount * (0.5)^(number of half-lives).
- Number of half-lives = total time / half-life period.
- Can be solved using logarithms or by recognizing powers of 0.5.
Memory trick: Halves 'H'appen in 'H'eaps: count the number of 'H'alves to find the age.