A drug decays by 29% an hour. Find the half-life using continuous exponential decay.

To determine the half-life of a drug that decays by 29% per hour, we can model this situation using continuous exponential decay. Here’s the step-by-step process:

1. **Define the decay rate**:
   The decay rate is 29%, so we can write the decay constant as \( k = -0.29 \) per hour.

2. **Set up the decay formula**:
   The continuous decay model is represented by:
   \[
   N(t) = N_0 e^{kt}
   \]
   where:
   - \( N(t) \) is the amount remaining after time \( t \),
   - \( N_0 \) is the initial amount,
   - \( k \) is the decay constant (here, \( -0.29 \) per hour).

3. **Define the half-life equation**:
   The half-life \( t_{1/2} \) is the time it takes for the substance to reduce to half its initial amount. Thus, we set \( N(t) = \frac{N_0}{2} \).

   \[
   \frac{N_0}{2} = N_0 e^{(-0.29)t_{1/2}}
   \]

4. **Solve for \( t_{1/2} \)**:
   Divide both sides by \( N_0 \):
   \[
   \frac{1}{2} = e^{(-0.29)t_{1/2}}
   \]

   Take the natural logarithm on both sides:
   \[
   \ln\left(\frac{1}{2}\right) = -0.29 \, t_{1/2}
   \]

   Since \( \ln\left(\frac{1}{2}\right) = -\ln(2) \), rewrite it as:
   \[
   -\ln(2) = -0.29 \, t_{1/2}
   \]

   Divide by \( -0.29 \) to isolate \( t_{1/2} \):
   \[
   t_{1/2} = \frac{\ln(2)}{0.29}
   \]

5. **Calculate the half-life**:
   Using \( \ln(2) \approx 0.693 \):
   \[
   t_{1/2} = \frac{0.693}{0.29} \approx 2.39 \text{ hours}
   \]

**Answer**: The half-life of the drug is approximately **2.39 hours**.

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### Additional Questions
1. What is the formula for calculating decay using different decay rates?
2. How would this decay model change for a compound that decays by a different percentage?
3. How does this half-life compare to that of substances in pharmacology?
4. What happens to the decay model if the drug is administered continuously?
5. Can this model apply to biological decay in living organisms?

**Tip**: The half-life formula for continuous decay, \( t_{1/2} = \frac{\ln(2)}{k} \), is very useful in fields like pharmacology and physics, where exponential decay is common.

Learn to calculate the half-life of a drug that decays at a rate of 29% per hour. This solution covers continuous exponential decay models, half-life formulas, and key mathematical concepts including logarithmic functions and natural logarithms. Ideal for advanced high school or college-level students.

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