Certainly! Let's analyze the problem step-by-step to determine how much of the initial 7 milligrams dose of Diazepam will remain in the patient's body after 3 days.
### Step 1: Understand the Decay Function
The decay function given is [tex]\( r(t) = 2.05 \times (0.83)^t \)[/tex]. However, we are interested in how much of the initial 7 mg dose remains after 3 days, rather than the rate of decay.
### Step 2: Calculate the Decay Rate after 3 Days
To calculate the decay rate specifically after 3 days, we evaluate the decay function at [tex]\( t = 3 \)[/tex]:
[tex]\[
r(3) = 2.05 \times (0.83)^3
\][/tex]
Evaluating [tex]\( (0.83)^3 \)[/tex]:
[tex]\[
(0.83)^3 \approx 0.571787
\][/tex]
Then,
[tex]\[
r(3) = 2.05 \times 0.571787 \approx 1.17216335
\][/tex]
This represents the factor by which the drug decays, meaning the rate contributed to the decay factor.
### Step 3: Calculate the Remaining Dose After 3 Days
We need to multiply the initial dose by the fraction of the dose that remains after 3 days. This fraction is given by [tex]\( (0.83)^3 \)[/tex]. So, we calculate:
[tex]\[
\text{Remaining dose} = \text{Initial dose} \times (0.83)^3
\][/tex]
[tex]\[
\text{Remaining dose} = 7 \times 0.571787 \approx 4.002509
\][/tex]
### Step 4: Round the Result to Two Decimal Places
To get the final answer in the required format (two decimal places):
[tex]\[
\text{Remaining dose} \approx 4.00 \text{ mg}
\][/tex]
### Conclusion
After 3 days, approximately 4.00 mg of the initial 7 mg dose of Diazepam will remain in the patient’s body (rounded to two decimal places).
We have reached this result by considering the decay rate and how it influences the remaining dose over the specified time period.
