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The problem involves constructing an exponential model for the decay of quinine in the body and then predicting how much remains after 24 hours. Here's the step-by-step solution:

1. **Exponential decay model**:
   The general form of the exponential decay model is:
   \[
   A(t) = A_0 e^{-kt}
   \]
   Where:
   - \( A(t) \) is the amount of quinine remaining after \( t \) hours,
   - \( A_0 \) is the initial dose of quinine,
   - \( k \) is the decay constant (which is related to the 6% per hour rate of decay),
   - \( t \) is the time in hours.

2. **Determine the decay constant \( k \)**:
   The problem states that quinine leaves the body at a rate of 6% per hour, which means 94% remains after each hour. Therefore, the decay rate per hour can be expressed as \( 1 - 0.06 = 0.94 \). The equation becomes:
   \[
   A(t) = A_0 (0.94)^t
   \]

3. **Substitute initial values**:
   Given \( A_0 = 165 \) mg (initial dose) and \( t = 24 \) hours, the model becomes:
   \[
   A(24) = 165 \times (0.94)^{24}
   \]

4. **Calculate the remaining quinine after 24 hours**:
   \[
   A(24) = 165 \times (0.94)^{24} \approx 165 \times 0.2359 \approx 38.93 \text{ mg}
   \]
   Rounded to the nearest tenth, the amount of quinine remaining after 24 hours is **38.9 mg**.

Let me know if you would like further details or have any questions. Here are some related questions to explore:

1. How would the model change if the decay rate was different, say 10% per hour?
2. What would the amount of quinine be after 48 hours?
3. How much quinine remains after just 12 hours?
4. How long will it take for only 50 mg of quinine to remain in the body?
5. What happens to the amount of quinine as \( t \to \infty \)?

**Tip**: Exponential decay processes are commonly used to model drug elimination, radioactive decay, and population decline. Understanding the decay constant is key to these problems.

A 165 mg dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of 6% per hour. Construct an exponential model to represent the number of milligrams of quinine in the body t hours after the dose is given. Use your model to predict the number of milligrams that will remain in the body after 24 hours.

Learn how to construct an exponential decay model for a 165 mg dose of quinine, which leaves the body at 6% per hour. Find out how much quinine remains after 24 hours using the formula A(t) = A0 * (0.94)^t. Suitable for Grades 9-12.

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