Haley conducted a study which found that a cup of coffee contains 150 milligrams of caffeine. The amount of caffeine in the body each hour after consumption of one cup is 9% less than the previous hour.
If Haley conducted her study for a total of 10 hours, which inequality represents the range of the exponential function that models this situation?
A. 0 <= f(x) <= 150
B. 0 <= f(x) <= 10
C. 58.41 <= f(x) <= 150
D. 150 <= f(x) <= 355.1
The correct answer and explanation is:
The correct answer is C. 58.41 <= f(x) <= 150.
Explanation:
This problem is about modeling the decay of caffeine in the body over time using an exponential function. The initial caffeine amount in the body is 150 milligrams, and every hour the caffeine level decreases by 9% compared to the previous hour. This suggests that the caffeine in the body follows an exponential decay function.
The general form for exponential decay is: f(x)=a⋅(1−r)xf(x) = a \cdot (1 – r)^x
Where:
- aa is the initial amount (150 mg of caffeine),
- rr is the rate of decay (9% or 0.09),
- xx is the time in hours, and
- f(x)f(x) is the amount of caffeine remaining after xx hours.
In this case, the exponential function becomes: f(x)=150⋅(0.91)xf(x) = 150 \cdot (0.91)^x
The caffeine decreases over time, but it never reaches zero because 9% of the caffeine remains each hour. Therefore, as xx increases, the caffeine level approaches but never reaches zero.
To find the caffeine level after 10 hours: f(10)=150⋅(0.91)10≈58.41 mgf(10) = 150 \cdot (0.91)^{10} \approx 58.41 \, \text{mg}
This shows that the caffeine level ranges from 150 mg (at time 0) to about 58.41 mg (after 10 hours).
The correct inequality representing the range of this exponential function is: 58.41≤f(x)≤15058.41 \leq f(x) \leq 150
Thus, the answer is C. 58.41 <= f(x) <= 150.