Answered

The amount of a radioactive substance remaining as it decays over time is $A=A_0(0.5)^{\frac{t}{h}}$, where $A$ represents the final amount, $A_0$ represents the original amount, $t$ represents the number of years, and $h$ represents the half-life of the substance.



Carbon-14 is a radioactive isotope that has a half-life of 5,730 years. Approximately how many years will it take for carbon-14 to decay to 10 percent of its original amount?


A16,396 years
B19,035 years
C28,650 years
D8,267 years

Verified Answer :

  • Use the radioactive decay formula \(A = A_0(0.5)^{\frac{t}{h}}\) with \(A = 0.10 A_0\) and \(h = 5730\).
  • Simplify the equation to \(0.10 = (0.5)^{\frac{t}{5730}}\).
  • Apply logarithms to isolate the time variable: \(t = 5730 \cdot \frac{\ln(0.10)}{\ln(0.5)}\).
  • Compute the final result, which is approximately \(\boxed{19{,}035}\) years.

Explanation

  1. Understanding the Decay Formula and the Problem

The problem asks us to determine the time \(t\) required for a sample of Carbon-14 to decay to \(10\%\) of its original amount. The radioactive decay is governed by the formula \(A = A_0(0.5)^{\frac{t}{h}}\), where \(A\) is the final amount, \(A_0\) is the initial amount, \(t\) is the time in years, and \(h\) is the half-life of the substance. For Carbon-14, the half-life \(h\) is \(5{,}730\) years. We want to find \(t\) such that \(A = 0.10 A_0\).

  1. Setting Up the Decay Equation

By substituting the known values \(A = 0.10 A_0\) and \(h = 5{,}730\) into the decay equation, we have:
\[0.10 A_0 = A_0(0.5)^{\frac{t}{5730}}.\]
Dividing both sides by the initial amount \(A_0\) (which is non-zero), we obtain the equation:
\[0.10 = (0.5)^{\frac{t}{5730}}.\]

  1. Solving for Time \(t\) Using Logarithms

To solve for \(t\), we take the natural logarithm (\(\ln\)) of both sides:
\[\ln(0.10) = \ln\left((0.5)^{\frac{t}{5730}}\right).\]
Using the logarithm property \(\ln(a^b) = b \cdot \ln(a)\), we can bring down the exponent:
\[\ln(0.10) = \frac{t}{5730} \cdot \ln(0.5).\]
Now, isolate \(t\) by multiplying both sides by \(5{,}730\) and dividing by \(\ln(0.5)\):
\[t = 5730 \cdot \frac{\ln(0.10)}{\ln(0.5)}.\]

  1. Calculating the Final Value and Selecting the Option

Using the calculated values for the natural logarithms, \(\ln(0.10) \approx -2.302585\) and \(\ln(0.5) \approx -0.693147\):
\[t = 5730 \cdot \frac{-2.302585}{-0.693147} \approx 5730 \cdot 3.321928 \approx 19{,}034.65.\]
Rounding to the nearest whole number, we get \(t \approx 19{,}035\) years. Comparing this to the given options, we identify the correct choice.

Examples

Carbon dating is a fundamental technique used in archaeology and geology to determine the age of organic materials. By measuring the remaining concentration of Carbon-14—which decays at a known, constant rate—scientists can backtrack to estimate when an organism died. For instance, if an ancient artifact is found to contain only 10% of the Carbon-14 levels expected in a living organism, the calculation performed above helps determine that the artifact is approximately 19,035 years old. This methodology serves as an essential chronological tool for understanding historical and evolutionary timelines.

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