1. **State the problem:** We are given that the remaining amount of carbon-14 in the cave paintings is 4% of the original amount. We want to estimate the age $t$ of the paintings using the exponential decay model. 2. **Formula:** The exponential decay model is given by: $$A = A_0 e^{-0.000121t}$$ where: - $A$ is the amount of carbon-14 remaining, - $A_0$ is the original amount of carbon-14, - $t$ is the time in years, - $0.000121$ is the decay constant. 3. **Set up the equation:** Since the paintings contain 4% of the original carbon-14, we have: $$\frac{A}{A_0} = 0.04 = e^{-0.000121t}$$ 4. **Solve for $t$:** Take the natural logarithm of both sides: $$\ln(0.04) = \ln\left(e^{-0.000121t}\right) = -0.000121t$$ 5. **Isolate $t$:** $$t = \frac{-\ln(0.04)}{0.000121}$$ 6. **Calculate $t$:** $$\ln(0.04) \approx -3.2189$$ $$t = \frac{-(-3.2189)}{0.000121} = \frac{3.2189}{0.000121} \approx 26611$$ 7. **Final answer:** The paintings are approximately **26611** years old (rounded to the nearest integer).