Problem 20 - Precalculus Mathematics for Calculus

Question

Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after \(t\) hours is modeled by $$D(t)=50 e^{-0.2 t}$$ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?

Step-by-Step Solution

Answer

Approximately 27.44 milligrams remain after 3 hours.

Step 1: Understanding the Function

The function given is \(D(t) = 50 e^{-0.2 t}\), which represents the number of milligrams of a drug remaining in the bloodstream after \(t\) hours. Our task is to find \(D(3)\) - the amount of drug remaining after 3 hours.

Step 2: Substitute t = 3 into the Function

To determine the drug amount after 3 hours, substitute \(t = 3\) into the equation: \(D(3) = 50 e^{-0.2 \times 3}\).

Step 3: Calculate the Exponent

Compute the exponent first: \(-0.2 \times 3 = -0.6\). So, the term becomes \(e^{-0.6}\).

Step 4: Evaluate the Exponential Term

Use a calculator to find \(e^{-0.6}\). This results in approximately 0.5488.

Step 5: Calculate the Remaining Drug Amount

Now, multiply the result from the exponential calculation with 50: \(50 \times 0.5488 \approx 27.44\).

Step 6: Interpret the Result

After completing the calculations, \(D(3)\approx 27.44\). This means approximately 27.44 milligrams of the drug remain after 3 hours.

Key Concepts

Exponential Functions

Exponential functions play a critical role in defining and understanding changes that occur at constant proportional rates. In mathematics, an exponential function is often expressed as \( f(x) = a \times e^{bx} \), where \( e \) is the base of the natural logarithm and \( a \) and \( b \) are constants.
In the context of decay, like in the original exercise where a drug's effectiveness decreases over time, the exponential function captures this decrease smoothly and predictably. The negative exponent \(-b\) indicates decay as opposed to growth. In the equation \( D(t) = 50 e^{-0.2t} \), the function shows an exponential decay. Here, 50 is the initial amount of the drug in milligrams. The exponent \(-0.2 \times t\) influences how quickly the drug concentration reduces over time. As \( t \) increases, the value of \( e^{-0.2t} \) decreases, leading to a decrease in \( D(t) \).
Exponential decay functions like this are powerful tools in various fields to model behaviors and predict future values.

Drug Dosage Calculations

Drug dosage calculations are essential in healthcare to ensure patients receive the right amount of medication over time. These calculations use mathematical models such as the exponential decay function above, to predict how drug dosages change within the body.
When a medication is administered, the body metabolizes it, leading to a decrease in concentration in the bloodstream. Knowing how fast or slow this rate is helps in planning subsequent doses to maintain therapeutic levels.
In the given model, \( D(t) = 50 e^{-0.2t} \), we determine the remaining drug amount in the body over different time periods. Calculating values like \( D(3) \) - 27.44 mg remaining after 3 hours - gives healthcare providers insight into its concentration.

• Proper timing between doses ensures safety and effectiveness.
• Understanding the exponential decay rate helps in adjusting doses appropriately.

Careful calculations are imperative to avoid underdosing, which reduces efficacy, and overdosing, which can be harmful.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to depict real-world phenomena. Exponential functions are commonly used in this modeling for scenarios representing growth and decay, like population growth, radioactive decay, and drug metabolism.
Our original exercise uses mathematical modeling through the function \( D(t) = 50 e^{-0.2t} \). It captures the essence of the biological process where a drug's quantity fades exponentially with time. These models help simulate various scenarios:

• Predicting drug concentration at given times post-ingestion.
• Guiding the creation of dosing schedules.
• Offering a graphical interpretation of how quickly drugs are metabolized.

These models must be precise and repeatedly validated against real-world data.
Relevance and accuracy in mathematical modeling enable effective decision-making and problem-solving. They offer insight into understanding complex processes simply, supporting application in diverse fields from medicine to environmental science.