To determine the time it takes for an IV drug with a half-life of 3 hours to reach approximately 98.5% of its steady-state concentration, we can use the following steps:
1. Understanding the half-life and steady-state concentration:
- The half-life of the drug is the time it takes for the concentration of the drug in the bloodstream to reduce by half.
- The steady-state concentration is the point at which the rate of drug administration equals the rate of drug elimination.
2. Determining the relevant variables:
- Half-life ([tex]\( t_{1/2} \)[/tex]) = 3 hours
- Desired steady-state percentage = 98.5%
3. Conversion to decimal for calculations:
- Steady-state fraction ([tex]\( \text{fraction} \)[/tex]) = 98.5%/100 = 0.985
4. Using the formula to find the time to reach a certain percentage of steady-state:
The formula to calculate the time to reach a specific percentage of the steady-state concentration for a drug with a given half-life is:
[tex]\[
t = \left( \frac{ t_{1/2} \cdot \ln \left( \frac{1}{1 - \text{fraction}} \right)}{\ln(2)} \right)
\][/tex]
Here:
- [tex]\( t \)[/tex] is the time to reach the desired fraction of steady-state concentration.
- [tex]\( t_{1/2} \)[/tex] is the half-life of the drug.
- [tex]\( \ln \)[/tex] represents the natural logarithm function.
- Fraction is the decimal representation of the desired steady-state percentage.
5. Plugging in the values:
[tex]\[
t = \left( \frac{3 \cdot \ln \left( \frac{1}{1 - 0.985} \right)}{\ln(2)} \right)
\][/tex]
6. Solving the equation:
[tex]\[
t \approx 18.176681067160704 \text{ hours}
\][/tex]
7. Rounding to the nearest whole number:
- Since the question requests the answer as a WHOLE number, we round 18.176681067160704 to the nearest whole number.
Thus, the time it takes for the drug to reach approximately 98.5% of its steady-state concentration is:
[tex]\[
\boxed{18}
\][/tex] hours.
