1. **State the problem:** We have a caffeine sample of 200 mg with a half-life of 5 hours. We want to find how much caffeine remains after 12 hours. 2. **Formula used:** The amount remaining after time $t$ is given by the exponential decay formula: $$ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} $$ where: - $A_0$ is the initial amount (200 mg), - $T$ is the half-life (5 hours), - $t$ is the elapsed time (12 hours). 3. **Substitute values:** $$ A = 200 \times \left(\frac{1}{2}\right)^{\frac{12}{5}} $$ 4. **Calculate the exponent:** $$ \frac{12}{5} = 2.4 $$ 5. **Evaluate the power:** $$ \left(\frac{1}{2}\right)^{2.4} = 2^{-2.4} $$ 6. **Calculate the remaining amount:** $$ A = 200 \times 2^{-2.4} $$ 7. **Approximate the value:** $$ 2^{-2.4} \approx 0.189 $$ $$ A \approx 200 \times 0.189 = 37.8 $$ **Final answer:** After 12 hours, approximately **37.8 mg** of caffeine remains.