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Difficulty: HardExponential Functions and Equations

The concentration of a medication in a patient's bloodstream, in milligrams per liter (mg/L\text{mg/L}), hh hours after injection is modeled by an exponential decay function. If the concentration of the medication decreases by 48.8%48.8\% every 33 hours, which of the following is closest to the percent decrease of the medication concentration each hour?

  1. A
    16.3%
  2. 20.0%Answer
  3. C
    80.0%
  4. D
    86.6%

Answer

20.0%
The correct answer is 20.0%. Since the concentration decreases by 48.8% every 3 hours, the remaining concentration after 3 hours is 51.2%, which corresponds to a factor of 0.512 of the initial concentration. The hourly decay factor bb satisfies the equation b3=0.512b^3 = 0.512. Taking the cube root of both sides yields b=0.8b = 0.8. A decay factor of 0.8 means that 80% of the medication remains each hour, representing a percent decrease of 20% each hour.

Step-by-Step Solution

1
Determine the remaining fraction of the medication after 3 hours.
Remaining fraction = 1 - 0.488 = 0.512
Since the concentration decreases by 48.8% every 3 hours, the remaining concentration is 51.2% of its value at the beginning of the 3-hour interval.
2
Set up the equation for the hourly retention factor, bb.
b3=0.512b^3 = 0.512
An exponential decay model of the form C(h)=C0⋅bhC(h) = C_0 \cdot b^h retains a factor of b3b^3 over a 3-hour period.
3
Solve for the hourly retention factor, bb.
b=0.5123=0.8b = \sqrt[3]{0.512} = 0.8
Taking the cube root of 0.512 gives the hourly factor, indicating that the concentration retains 80% of its value each hour.
4
Calculate the hourly percent decrease from the hourly retention factor.
Percent decrease = (1−0.8)×100%=20.0%(1 - 0.8) \times 100\% = 20.0\%
The hourly percent decrease is the difference between the initial 100% and the remaining 80% retained each hour.

Key Concept

Converting decay factors across different time intervals using exponent properties.
Estimated Time:2m 0s
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Topics

Exponential Functions and Equations

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