Problem 11 - Applied Calculus

Question

A 50 mg dose of quinine is given to a patient to prevent malaria. Quinine leaves the body at a rate of \(6 \%\) per hour. (a) Find a formula for the amount, \(A\) (in \(\mathrm{mg}\) ), of quinine in the body \(t\) hours after the dose is given. (b) How much quinine is in the body after 24 hours? (c) Graph \(A\) as a function of \(t\) (d) Use the graph to estimate when 5 mg of quinine remains.

Step-by-Step Solution

Answer

(a) Formula: \( A(t) = 50 \times 0.94^t \) (b) 14.31 mg (c) Use a graphing tool to plot (d) About 50 hours.

Step 1: Define the Problem

We need to determine how quinine exits the body over time and develop a mathematical model for this process.

Step 2: Establish the Rate Function

Quinine leaves the body at a rate of 6% per hour. This can be represented by the exponential decay formula, where the rate of decay is 6% per hour.

Step 3: Derive the Exponential Decay Formula

The general formula for exponential decay is given by \( A(t) = A_0 \times (1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is time in hours. Here, \( A_0 = 50 \) mg and \( r = 0.06 \). Thus, the formula becomes \( A(t) = 50 \times 0.94^t \).

Step 4: Calculate Amount After 24 Hours

To find the amount of quinine after 24 hours, substitute \( t = 24 \) into the formula: \( A(24) = 50 \times 0.94^{24} \). Calculate to find the result.

Step 5: Graph the Function

Create a plot of the function \( A(t) = 50 \times 0.94^t \) to visualize how the amount of quinine decreases over time. Use a graphing tool to draw the graph with \( t \) on the x-axis and \( A(t) \) on the y-axis.

Step 6: Estimate Time for 5 mg

Using the graph, find the point where the amount of quinine is approximately 5 mg. This corresponds to when \( A(t) = 5 \).

Key Concepts

Calculating Medicinal Dosage

Medicinal dosage calculations, especially when dealing with drugs like quinine, are crucial for ensuring safe and effective treatment. In our example, we start with a 50 mg dose of quinine, given to battle malaria. The key challenge is to figure out how much of this drug remains in the body over time. This is due to the body's natural process of eliminating substances, which for quinine happens at a rate of 6% per hour. To determine the remaining dosage, we use the formula for exponential decay: \[ A(t) = A_0 \times (1 - r)^t \] Here:

• \(A_0\) represents the initial dose, which is 50 mg.
• \(r\) is the decay rate, 6% (or 0.06 in decimal).
• \(t\) denotes time in hours.

Plugging in our values gives us \( A(t) = 50 \times 0.94^t \). This formula allows us to calculate how much quinine is left in the body after any number of hours.

Graphing Exponential Functions

Visualizing how quinine levels decrease can be very insightful. We can achieve this through graphing. By plotting the function \( A(t) = 50 \times 0.94^t \), we create a visual representation of the drug's presence over time. In this graph, the horizontal axis (x-axis) marks time in hours, while the vertical axis (y-axis) shows the amount of quinine in mg. As time progresses, you should notice a downward slope, characteristic of exponential decay, reflecting how the drug diminishes.Graphing helps in:

• Checking your calculations visually.
• Estimating key points, such as when only a specific amount (e.g., 5 mg) remains.
• Understanding the practical implications of dosage schedules.

To determine when exactly 5 mg remains, one can look at where the graph crosses the y-value of 5 mg.

Mathematical Modeling in Medicine

Mathematical modeling plays a vital role in healthcare, particularly in drug dosing. By employing models like exponential decay, healthcare professionals can predict how drugs metabolize within the body. This aids in creating effective dosing schedules, minimizing side effects, and ensuring the medication remains at therapeutic levels. In medical modeling, understanding how decay functions operate is essential because:

• They provide precise quantifiable data over time.
• These models simplify complex biological processes into understandable patterns.
• Accurate predictions help in designing treatment plans for different patients based on their metabolic rates.

For quinine, modeling helps ensure that the levels stay effective to combat malaria without the need for frequent dosing, ultimately improving treatment adherence and outcomes.