To model the mass [tex]\( M(t) \)[/tex] of a Carbon-14 sample remaining after [tex]\( t \)[/tex] years since the initial measurement, we need to consider that the sample loses 10% of its mass every 871 years. This indicates that for every 871-year period, the sample retains 90% of its mass.
Let's break down the steps to derive the function [tex]\( M(t) \)[/tex]:
1. Initial Mass and Decay Rate:
- Initial mass of the sample: 960 grams.
- Percentage loss per 871 years: 10%.
- Mass retention per 871 years: Thus, it retains 90% of its mass, which is equivalent to a factor of 0.90.
2. Number of Decay Periods:
- Determine how many 871-year periods are contained in [tex]\( t \)[/tex] years. This can be calculated as [tex]\( \frac{t}{871} \)[/tex].
3. Exponential Decay Model:
- After each period of 871 years, the remaining mass can be expressed as [tex]\( 0.90 \)[/tex] times the mass before that period.
- Thus, after [tex]\( t \)[/tex] years, the mass [tex]\( M(t) \)[/tex] can be modeled by raising the retention factor to the power of the number of periods [tex]\( \frac{t}{871} \)[/tex].
4. Combining the Information:
- The general formula for the mass remaining after [tex]\( t \)[/tex] years is:
[tex]\[
M(t) = \text{initial mass} \times (\text{retention rate})^{\text{number of periods}}
\][/tex]
- Substituting the specific values:
[tex]\[
M(t) = 960 \times 0.90^{\frac{t}{871}}
\][/tex]
Therefore, the function [tex]\( M(t) \)[/tex] that models the mass of the carbon-14 sample remaining after [tex]\( t \)[/tex] years is:
[tex]\[
M(t) = 960 \times 0.90^{\frac{t}{871}}
\][/tex]
This function correctly captures the exponential decay behavior of the carbon-14 sample over time.
