prehistoric cave paintings were discovered in a cave in france. the paint contained 12% of the original…

prehistoric cave paintings were discovered in a cave in france. the paint contained 12% of the original carbon - 14. use the exponential decay model for carbon - 14, ( a = a_0e^{-0.000121t} ), to estimate the age of the paintings. the paintings are approximately ( square ) years old. (round to the nearest integer.)

prehistoric cave paintings were discovered in a cave in france. the paint contained 12% of the original carbon - 14. use the exponential decay model for carbon - 14, ( a = a_0e^{-0.000121t} ), to estimate the age of the paintings. the paintings are approximately ( square ) years old. (round to the nearest integer.)

Answer

Explanation:

Step1: Set up the equation

Given the formula (A = A_0e^{- 0.000121t}), and since the paint contains (12%) of the original carbon - 14, we have (A=0.12A_0). Substitute (A = 0.12A_0) into the formula: (0.12A_0=A_0e^{-0.000121t}) Divide both sides by (A_0) (since (A_0\neq0)), we get (0.12 = e^{-0.000121t})

Step2: Take the natural logarithm of both sides

Using the property (\ln(e^{x})=x), take the natural logarithm of both sides of the equation (0.12 = e^{-0.000121t}). (\ln(0.12)=\ln(e^{-0.000121t})) Since (\ln(e^{-0.000121t})=- 0.000121t), our equation becomes (\ln(0.12)=-0.000121t)

Step3: Solve for (t)

We know that (\ln(0.12)\approx - 2.1203). Then, (t=\frac{\ln(0.12)}{-0.000121}) Substitute (\ln(0.12)\approx - 2.1203) into the formula: (t=\frac{-2.1203}{-0.000121}) (t\approx17523.14)

Answer:

(17523)