The amount of caffeine decreases by 9% each hour, meaning it retains 91% of the caffeine from the previous hour. The function modeling this is \(f( x) = 150 \cdot ( 0.91) ^ x\). After 10 hours, \(f( 10) = 150 \cdot ( 0.91) ^ { 10} \approx 58.41\). Thus, the range of \(f( x) \) over 10 hours is from 58.41 to 150 milligrams.

Supplemental Knowledge

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This is often used to model situations like radioactive decay, depreciation of assets, and the reduction of substances in the body over time.
The general form of an exponential decay function is:
\[f( t) = A \cdot ( 1 - r) ^ t\]
where:

• \(A\) is the initial amount.
• \(r\) is the decay rate (as a decimal).
• \(t\) is the time period.
  In this context, Haley's study found that caffeine decreases by 9% each hour. Therefore, \(r = 0.09\), and the function modeling the caffeine amount in the body after \(t\) hours can be written as:
  \[f( t) = 150 \cdot ( 1 - 0.09) ^ t\]
  or equivalently,
  \[f( t) = 150 \cdot 0.91^ t\]

From Concepts to Reality

Consider how your body processes caffeine when you drink coffee. Initially, you feel alert and energized, but over time, your body metabolizes the caffeine, reducing its concentration in your bloodstream. This gradual decrease follows an exponential decay pattern similar to how Haley's study describes it.
For example, if you drink a cup of coffee at 8 AM with 150 milligrams of caffeine, by 9 AM (one hour later), you'd have approximately \(150 \times 0.91\) milligrams left in your system. By understanding this pattern, you can better manage your caffeine intake throughout the day to avoid late-night jitters or ensure sustained energy levels.

Grasping exponential decay can provide valuable insights into various real-world phenomena such as pharmacokinetics and financial depreciation. At UpStudy, we offer advanced tools like our Algebra functions calculator to help you master these concepts effortlessly. Dive into our extensive resources and live tutor question bank for personalized guidance on mathematical modeling and more.
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