To solve the problem of determining how many milligrams of the drug will remain in the patient's body after 9 days, given the drug's half-life and initial amount, we should use the formula for exponential decay based on half-life:
[tex]\[ R = A \left(\frac{1}{2}\right)^{\frac{t}{T}} \][/tex]
where:
- [tex]\( R \)[/tex] is the remaining amount of the drug.
- [tex]\( A \)[/tex] is the initial amount of the drug.
- [tex]\( t \)[/tex] is the time elapsed.
- [tex]\( T \)[/tex] is the half-life period of the drug.
Given:
- Initial amount [tex]\( A = 4 \)[/tex] milligrams.
- Half-life [tex]\( T = 3 \)[/tex] days.
- Time elapsed [tex]\( t = 9 \)[/tex] days.
Let's plug these values into our formula:
[tex]\[ R = 4 \left(\frac{1}{2}\right)^{\frac{9}{3}} \][/tex]
First, compute the exponent:
[tex]\[ \frac{9}{3} = 3 \][/tex]
Thus, the formula simplifies to:
[tex]\[ R = 4 \left(\frac{1}{2}\right)^3 \][/tex]
Now, calculate [tex]\( \left(\frac{1}{2}\right)^3 \)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \][/tex]
So we have:
[tex]\[ R = 4 \times \frac{1}{8} \][/tex]
Finally, perform the multiplication:
[tex]\[ R = \frac{4}{8} = 0.5 \][/tex]
Therefore, the remaining amount of the drug in the patient's body after 9 days will be [tex]\(\boxed{0.5}\)[/tex] milligrams.
