QuestionPrehistoric cave paintings were discovered in a cave in France. The paint contained $32 \%$ of the original carbon-14. Use the exponential decay model for carbon-14, $A=A_{0} e^{-0.000121 t}$, to estimate the age of the paintings.
The paintings are approximately $\square$ years old. (Round to the nearest integer.)

Studdy Solution
Solve for the variable $t$.
First, divide both sides by $A_{0}$ to isolate the exponential term:
$$0.32 = e^{-0.000121 t}$$
Take the natural logarithm of both sides to solve for $t$:
$$\ln(0.32) = \ln(e^{-0.000121 t})$$
Using the property of logarithms, simplify the right side:
$$\ln(0.32) = -0.000121 t$$
Now, solve for $t$ by dividing both sides by $-0.000121$:
$$t = \frac{\ln(0.32)}{-0.000121}$$
Calculate the value:
$$t \approx \frac{-1.139434}{-0.000121}$$
$$t \approx 9415.16$$
Round to the nearest integer:
The paintings are approximately $\boxed{9415}$ years old.