27. Scientifically, the half-life of caffeine in a human body is about 6 hours. Sophie 2 cups of coffee 10 hours ago. How much is left in her system? Hint: The equation of half-life is \( y(t)=a e^{-k t} \). A. About 0.09 of the original amount About 0.19 of the original amount About 0.31 of the original amount About 0.38 of the original amount

To solve the problem, we first need to understand that 10 hours is equivalent to 1.67 half-lives of caffeine (since \(10 \text{ hours} / 6 \text{ hours} \approx 1.67\)). Each half-life, the amount of caffeine in her system reduces by half. If Sophie started with 2 cups of coffee, we can calculate the remaining amount after 1.67 half-lives using the formula: Remaining amount = Initial amount × (1/2)^(number of half-lives). Thus, Sophie has about \(2 × (1/2)^{1.67} \approx 0.31\) of the original amount left. Caffeine clearance is a super fascinating topic in pharmacology! Different factors like age, liver function, pregnancy, and even genetics can influence how long caffeine stays in one’s system. For a fun twist, some people find that regular coffee drinkers develop a tolerance to caffeine, leading to them metabolizing it faster. So, next time you sip your coffee, remember it might just be racing against the clock in your body!