1. **State the problem:** We have a caffeine sample of 200 mg with a half-life of 5 hours. We want to find out how much caffeine remains after 12 hours. 2. **Formula used:** The amount remaining after time $t$ is given by the formula for exponential decay: $$ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} $$ where: - $A_0$ is the initial amount (200 mg), - $T$ is the half-life (5 hours), - $t$ is the elapsed time (12 hours), - $A$ is the amount remaining after time $t$. 3. **Plug in the 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:** $$ \left(\frac{1}{2}\right)^{2.4} = \frac{1}{2^{2.4}} $$ 6. **Calculate $2^{2.4}$:** $$ 2^{2.4} = 2^{2 + 0.4} = 2^2 \times 2^{0.4} = 4 \times 2^{0.4} $$ 7. **Approximate $2^{0.4}$:** Using a calculator or approximation, $2^{0.4} \approx 1.3195$ 8. **Multiply:** $$ 4 \times 1.3195 = 5.278 $$ 9. **Calculate the remaining amount:** $$ A = 200 \times \frac{1}{5.278} = 200 \times 0.1894 = 37.88 $$ 10. **Final answer:** Approximately **37.88 mg** of caffeine remains after 12 hours. This means after 12 hours, less than 20% of the original caffeine is left because it decays by half every 5 hours.