To model the decay of Carbon-14 over time, we can use the formula for exponential decay. Given that Carbon-14 loses approximately 10% of its mass every millennium, we can express the remaining mass after \( t \) millennia with the following formula:
- Initial mass \( S(0) = 600 \) grams
- Decay rate \( r = 10% = 0.10 \)
The remaining mass after \( t \) millennia can be calculated as:
\[ S(t) = S(0) \times (1 - r)^t \]
Substituting the values we have:
- \( S(0) = 600 \)
- \( r = 0.10 \)
The equation becomes:
\[ S(t) = 600 \times (1 - 0.10)^t \] \[ S(t) = 600 \times (0.90)^t \]
Thus, the final function that gives the sample's mass in grams \( S(t) \) after \( t \) millennia is:
\[ S(t) = 600 \times (0.90)^t \]
This function will provide the mass of the Carbon-14 sample at any given time \( t \) in millennia, reflecting the decay in mass over time.
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