To model the exponential decay of ibuprofen in an adult's system, we need a function that relates the amount of ibuprofen remaining, [tex]\( m(t) \)[/tex], to the time in hours, [tex]\( t \)[/tex].
Let's break down the problem step-by-step:
1. Initial Dose: The initial amount of ibuprofen taken is 400 milligrams (mg). This means at [tex]\( t = 0 \)[/tex] (the moment the adult takes the dose), [tex]\( m(0) = 400 \)[/tex].
2. Decay Rate: Each hour, the amount of ibuprofen decreases by one-fourth. This implies that after each hour, only [tex]\( \frac{1}{4} \)[/tex] (or 25%) of the previous amount remains. Therefore, we can conclude that the decay rate (fraction of the amount that remains from hour to hour) is [tex]\( \frac{1}{4} \)[/tex].
3. Exponential Function Form: Since this is an exponential decay problem, the general form of the function is:
[tex]\[
m(t) = \text{initial amount} \times (\text{decay rate})^t
\][/tex]
4. Substituting Values: The initial amount is 400 mg, and the decay rate is [tex]\( \frac{1}{4} \)[/tex]. Placing these values into the general form, we get:
[tex]\[
m(t) = 400 \times \left(\frac{1}{4}\right)^t
\][/tex]
Therefore, the function that models the exponential decay of ibuprofen in the person's system is:
[tex]\[
m(t) = 400 \cdot \left(\frac{1}{4}\right)^t
\][/tex]
So the correct function among the given options is:
[tex]\[
m(t) = 400 \cdot \left(\frac{1}{4}\right)^t
\][/tex]
