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.

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!