Question 3a of 4

If a prescription drug's half-life is 3 days and a patient currently has 4 milligrams of the drug in his system, how many milligrams of the drug will be in his body after 9 days?

Use the formula: [tex]R = A\left(\frac{1}{2}\right)^{\frac{t}{n}}[/tex]

A. [tex]\frac{1}{16}[/tex]
B. 32
C. [tex]\frac{1}{2}[/tex]
D. 8



Answer :

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.