Question

Carbon-
\[14\] is an element that loses about
\[10\%\] of its mass every millennium (i.e.,
\[1000\] years). A sample of Carbon-
\[14\] has
\[600\] grams.
Write a function that gives the sample's mass in grams,
\[S(t)\],
\[t\] millennia from today.
\[S(t)=\]

Answer 1

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:

1. **Initial mass** \( S(0) = 600 \) grams
2. **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.