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 6946 years.
the amount of carbon - 14 present in 6946 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 \(t = 6946\). Substitute \(t\) into the formula: \(A=16e^{-0.000121\times6946}\).
First, calculate the exponent: \(-0.000121\times6946=- 0.840466\).
So the formula becomes \(A = 16e^{-0.840466}\).

Step2: Calculate the value of \(e^{-0.840466}\)

We know that \(y = e^{x}\), when \(x=-0.840466\), \(e^{-0.840466}=\frac{1}{e^{0.840466}}\). Using a calculator, \(e^{0.840466}\approx2.317\), so \(e^{-0.840466}\approx\frac{1}{2.317}\approx0.4316\).

Step3: Calculate the value of \(A\)

Now, \(A = 16\times0.4316\). \(16\times0.4316 = 6.9056\).

Answer:

\(7\)