3. The function $$ M(t)=975 \cdot 0.92^{\prime} $$ represents the number of milligrams of a medication in a patient's body as a function of time.

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It appears you're presenting a function that models the amount of medication in a patient's body over time. Let's clarify and analyze the function you've provided:

$$
M(t) = 975 \cdot 0.92^t
$$

**Assumptions:**
1. **Initial Amount ($$ M(0) $$)**: At time $$ t = 0 $$, the amount of medication in the patient's body is $$ 975 $$ milligrams (mg).
   
   $$
   M(0) = 975 \cdot 0.92^0 = 975 \cdot 1 = 975 \, \text{mg}
   $$

2. **Decay Factor ($$ 0.92 $$)**: The base of the exponent, $$ 0.92 $$, represents the retention rate of the medication per unit time. This means that **each unit of time**, $$ 8\% $$ of the medication is metabolized or eliminated from the body, retaining $$ 92\% $$ of the previous amount.

3. **Time Variable ($$ t $$)**: The variable $$ t $$ typically represents time. The exact unit (e.g., hours, days) should be specified based on the context of the problem.

**Interpreting the Function:**

- **Exponential Decay Model**: The function $$ M(t) = 975 \cdot 0.92^t $$ is an example of an exponential decay model. It describes a situation where the quantity decreases by a constant percentage over equal intervals of time.

- **General Behavior**:
  - As $$ t $$ increases, $$ M(t) $$ decreases.
  - The medication amount never becomes negative but approaches zero asymptotically.
  
- **Half-Life Calculation**: If you're interested in finding out how long it takes for the medication amount to reduce to half its initial value, you can calculate the **half-life** ($$ t_{1/2} $$) using the formula:

$$
  M(t_{1/2}) = \frac{M(0)}{2} \Rightarrow 975 \cdot 0.92^{t_{1/2}} = \frac{975}{2}
  $$
  
  $$
  0.92^{t_{1/2}} = 0.5
  $$
  
  Taking the natural logarithm of both sides:
  
  $$
  \ln(0.92^{t_{1/2}}) = \ln(0.5)
  $$
  
  $$
  t_{1/2} \cdot \ln(0.92) = \ln(0.5)
  $$
  
  $$
  t_{1/2} = \frac{\ln(0.5)}{\ln(0.92)} \approx \frac{-0.6931}{-0.0834} \approx 8.3 \, \text{units of time}
  $$
  
  So, the half-life of the medication is approximately $$ 8.3 $$ time units (e.g., hours or days, depending on the context).

**Practical Implications:**

- **Dosing Schedule**: Understanding the decay rate helps healthcare providers determine how frequently a medication should be administered to maintain its therapeutic level without causing toxicity.

- **Patient Safety**: Knowing how long a medication stays in the body ensures that dosages are safe and effective, preventing potential side effects from accumulation or ineffectiveness due to rapid clearance.

**Graphical Representation:**

Plotting $$ M(t) = 975 \cdot 0.92^t $$ will yield a curve that starts at $$ 975 $$ mg when $$ t = 0 $$ and decreases exponentially towards zero as $$ t $$ increases. Here's a rough sketch of the behavior:

```
M(t)
|
|\
| \
|  \
|   \
|    \
|     \
|      \
|_______\________ t
```

**Key Points on the Graph:**
- At $$ t = 0 $$, $$ M(t) = 975 $$ mg.
- As $$ t $$ approaches infinity, $$ M(t) $$ approaches $$ 0 $$ mg.
- The curve is always decreasing but never touches the $$ t $$-axis.

**Conclusion:**

The function $$ M(t) = 975 \cdot 0.92^t $$ effectively models the exponential decay of a medication in a patient's body, allowing for the analysis of dosage schedules, half-life calculations, and ensuring patient safety through appropriate medication management.

If you have a specific question or a particular aspect of this model you'd like to explore further, please provide more details!
The function $$ M(t) = 975 \cdot 0.92^t $$ models the amount of medication in a patient's body over time, decreasing by 8% each unit of time. It starts at 975 mg and approaches zero as time increases. The half-life of the medication is approximately 8.3 time units.

3. The function \( M(t)=975 \cdot 0.92^{\prime} \) represents the number of milligrams of a medication in a patient's body as a function of time.

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