A painkiller has a concentration of 56 mg/deciliter 2.5 hours after it has been administered.  At 7 hours the concentration has fallen to 36 mg/deciliter.Find the rate of decay, the initial concentration, the exponential decay model, and the half life. 

To solve this problem, we will use the exponential decay model, which is commonly used to describe the decay of substances over time. The general formula for exponential decay is:

\[ C(t) = C_0 e^{-kt} \]

Where:
- \( C(t) \) is the concentration at time \( t \),
- \( C_0 \) is the initial concentration (at time \( t = 0 \)),
- \( k \) is the decay constant, and
- \( t \) is time.

Given:
- \( C(2.5) = 56 \) mg/dL (concentration after 2.5 hours)
- \( C(7) = 36 \) mg/dL (concentration after 7 hours)

### 1. Finding the Decay Constant (\( k \))

To find \( k \), we can use the concentrations at two different times. We have:

\[ 56 = C_0 e^{-2.5k} \]
\[ 36 = C_0 e^{-7k} \]

We can divide these equations to eliminate \( C_0 \):

\[ \frac{56}{36} = \frac{C_0 e^{-2.5k}}{C_0 e^{-7k}} \]

\[ \frac{56}{36} = e^{7k - 2.5k} \]

\[ \frac{56}{36} = e^{4.5k} \]

\[ \frac{56}{36} = e^{4.5k} \]

\[ 1.5556 = e^{4.5k} \]

\[ \ln(1.5556) = 4.5k \]

\[ k = \frac{\ln(1.5556)}{4.5} \]

Now, we calculate the value of \( k \):

\[ k \approx \frac{0.4417}{4.5} \]

\[ k \approx 0.0982 \, \text{per hour} \]

### 2. Finding the Initial Concentration (\( C_0 \))

Using \( k \) and one of the equations, we can find \( C_0 \):

\[ 56 = C_0 e^{-2.5 \times 0.0982} \]

\[ C_0 = \frac{56}{e^{-0.2455}} \]

\[ C_0 = \frac{56}{0.7824} \]

\[ C_0 \approx 71.6 \, \text{mg/dL} \]

### 3. Exponential Decay Model

The model can now be written as:

\[ C(t) = 71.6 e^{-0.0982t} \]

### 4. Finding the Half-Life

The half-life \( t_{1/2} \) is the time required for the concentration to reduce to half of its initial value. We use the equation:

\[ \frac{1}{2}C_0 = C_0 e^{-kt_{1/2}} \]

\[ \frac{1}{2} = e^{-kt_{1/2}} \]

\[ \ln\left(\frac{1}{2}\right) = -kt_{1/2} \]

\[ -\ln(2) = -kt_{1/2} \]

\[ t_{1/2} = \frac{\ln(2)}{k} \]

\[ t_{1/2} = \frac{\ln(2)}{0.0982} \]

\[ t_{1/2} \approx 7.05 \, \text{hours} \]

### Summary

- **Decay constant \( k \)**: \( 0.0982 \) per hour
- **Initial concentration \( C_0 \)**: \( 71.6 \) mg/dL
- **Exponential decay model**: \( C(t) = 71.6 e^{-0.0982t} \)
- **Half-life \( t_{1/2} \)**: \( 7.05 \) hours

Would you like further details on any of these calculations or have any additional questions?

Here are some questions you might consider:

1. How do you derive the exponential decay formula?
2. What are some real-world applications of exponential decay?
3. How is the half-life concept applied in pharmacology?
4. Can the rate of decay be influenced by external factors?
5. How would you graph the decay of the concentration over time?

**Tip:** Understanding the derivation of the exponential decay formula can help in applying the concept to various real-world scenarios, such as pharmacokinetics and radioactive decay.

Learn how to calculate exponential decay rate, initial concentration, and half-life of a substance using the exponential decay model. This problem involves detailed calculations and formulas to determine key pharmacokinetic parameters. Suitable for advanced high school students and beyond.

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