Question

an artifact originally had 16 grams of carbon-14 present. the decay model $a = 16e^{-0.0001211t}$ 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 7701 years. (round to the nearest whole number.) the amount of carbon-14 present in 7701 years will be approximately \boxed{} grams.

Explanation:

Step1: Identify the decay model formula

The decay model is given as \( A = 16e^{-0.0001211t} \), where \( t = 7701 \) years.

Step2: Substitute \( t = 7701 \) into the formula

Substitute \( t = 7701 \) into the formula: \( A = 16e^{-0.0001211\times7701} \)

Step3: Calculate the exponent

First, calculate the exponent: \( -0.0001211\times7701 \approx -0.9326 \)

Step4: Calculate the exponential term

Then, calculate \( e^{-0.9326} \approx 0.393 \)

Step5: Calculate the final amount

Multiply by 16: \( A = 16\times0.393 \approx 6.288 \)

Step6: Round to the nearest whole number

Rounding \( 6.288 \) to the nearest whole number gives 6.

Answer:

6