Question

an artifact originally had 16 grams of carbon - 14 present. the decay model ( a = 16e^{-0.000121t} ) describes the amount of carbon - 14 present after t years. use the model to determine how many grams of carbon - 14 will be present in 6156 years.
the amount of carbon - 14 present in 6156 years will be approximately (square) grams.
(round to the nearest whole number.)

Explanation:

Step1: Substitute \(t = 6156\) into the formula

We have the formula \(A=16e^{- 0.000121t}\). Substitute \(t = 6156\) into it: \(A = 16e^{-0.000121\times6156}\).
First, calculate the exponent: \(-0.000121\times6156=- 0.744876\).
So the formula becomes \(A = 16e^{-0.744876}\).

Step2: Calculate the value of \(e^{-0.744876}\)

We know that \(y = e^{x}\), and when \(x=-0.744876\), \(e^{-0.744876}=\frac{1}{e^{0.744876}}\). Using a calculator, \(e^{0.744876}\approx2.106\), so \(e^{-0.744876}\approx\frac{1}{2.106}\approx0.475\).

Step3: Calculate the value of \(A\)

Now, \(A = 16\times0.475\). \(16\times0.475 = 7.6\).

Answer:

\(8\)