Final answer:
Jan's body will have 28% of the original caffeine amount after approximately 4 hours. This is determined by applying the formula for exponential decay and solving for the time it takes for the caffeine to reduce to 28% of its original amount.
Step-by-step explanation:
The question deals with an exponential decay problem where the amount of caffeine in Jan's body decreases by 28% every hour. To solve this, we use the formula for exponential decay given by A = P(1 - r)^t, where A is the amount remaining, P is the initial amount, r is the decay rate, and t is the time in hours.
We want to find out when Jan's body has only 28% of the original caffeine amount. Setting A to 0.28P (since 28% is left), and r to 0.28 (since 28% is lost each hour), we can solve for t. The equation is now 0.28P = P(1 - 0.28)^t. Dividing both sides by P and simplifying gives us 0.28 = (0.72)^t.
Now we need to solve for t using logarithms or by trial and error. By checking the first few whole number powers of 0.72, we find:
- (0.72)^1 = 0.72
- (0.72)^2 = 0.5184
- (0.72)^3 = 0.373248
- (0.72)^4 = 0.26873856
The value (0.72)^4 is the closest to 0.28 without going over, thus it takes approximately 4 hours to reach 28% of the original caffeine amount. So the answer to the student's question is (c) 4 hours.