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An archaeologist finds a bone that he thinks dates back to the time of the pharaohs, about 14,000 years ago. A scientist has measured the C14 in the bone and finds that it is $\frac{1}{4}$ of the concentration in the atmosphere. Could the bone be as old as the archaeologist thinks?



The equation is $N(t)=0.5^m$.

The ordered pair for the archaeologist's conjecture is $(14,000, 0.25)$.



Does the ordered pair fit the equation?

Enter values in the boxes.

$N($ [ ] $) = 0.5^{[ ]}$

Verified Answer :

  • We evaluate the conjecture by substituting \(t = 14,000\) into the model \(N(t) = 0.5^t\), showing \(0.5^{14,000} eq 0.25\).
  • We conclude that the ordered pair \((14,000, 0.25)\) does not satisfy the equation.
  • We solve \(N(t) = 0.5^t = 0.5\) to find the time \(t\) that results in a concentration of 0.5, which is \(t = 1\).
  • The completed expression is \(N(\boxed{1}) = 0.5\).

Explanation

  1. Problem Analysis

We are analyzing an archaeologist's hypothesis that a bone is 14,000 years old, given the model \(N(t) = 0.5^t\). We are asked to test if the ordered pair \((14,000, 0.25)\) satisfies this equation and to determine the value of \(t\) that satisfies \(N(t) = 0.5\). The function \(N(t) = 0.5^t\) describes the remaining fraction of a substance after \(t\) time units, where in this specific model, one unit represents one half-life period.

  1. Evaluating the Conjecture

To determine if the ordered pair \((14,000, 0.25)\) fits the model \(N(t) = 0.5^t\), we substitute \(t = 14,000\) into the equation:
\[N(14,000) = 0.5^{14,000}\]
Comparing this to the value \(0.25\) (which is equal to \(0.5^2\)), it is clear that \(0.5^{14,000} eq 0.5^2\). Therefore, the ordered pair \((14,000, 0.25)\) does not satisfy the given equation \(N(t) = 0.5^t\). The archaeologist's conjecture is not consistent with this mathematical model.

  1. Solving for the Missing Value

To fill in the blank for the expression \(N(\Box) = 0.5\), we set the model equation equal to \(0.5\):
\[0.5^t = 0.5\]
Since \(0.5 = 0.5^1\), we have:
\[0.5^t = 0.5^1\]
By equating the exponents, we find that \(t = 1\). Thus, \(N(1) = 0.5\).

  1. Conclusion

The ordered pair \((14,000, 0.25)\) does not fit the equation \(N(t) = 0.5^t\), as \(N(14,000) = 0.5^{14,000} eq 0.25\). The value required to satisfy \(N(\Box) = 0.5\) is \(1\), so we have \(N(1) = 0.5\).

Examples

This problem demonstrates how mathematical models are used to verify archaeological hypotheses. By comparing experimental data (the measured concentration) against a theoretical decay model, scientists can determine if a proposed age is plausible or if the data contradicts the model. This is the foundation of radiocarbon dating, which helps establish reliable timelines for historical artifacts and ancient civilizations.

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