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 8578 years. the amount of carbon - 14 present in 8578 years will be approximately (square) grams. (round to the nearest whole number.)

Explanation:

Step1: Substitute the value of \(t\) into the formula

Given the formula \(A = 16e^{-0.000121t}\), and \(t = 8578\). Substitute \(t\) into the formula: \(A=16e^{-0.000121\times8578}\).
First, calculate the exponent: \(-0.000121\times8578=-1.038\) (approx).

Step2: Calculate the value of \(e^{-1.038}\)

We know that \(y = e^{x}\), when \(x=-1.038\), \(e^{-1.038}=\frac{1}{e^{1.038}}\). Using a calculator, \(e^{1.038}\approx2.824\), so \(e^{-1.038}\approx\frac{1}{2.824}\approx0.354\).

Step3: Calculate the value of \(A\)

Now, \(A = 16\times0.354 = 5.664\approx6\) (rounded to the nearest whole number).

Answer:

\(6\)