Problem 64 - Algebra and Trigonometry

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: Identify Variables and Function

The function given is \( D(t) = 50 e^{-0.2t} \), where \( D(t) \) represents the milligrams of the drug remaining in the bloodstream after \( t \) hours. We are asked to find \( D(3) \), the amount left after 3 hours.

Step 2: Substitute Variable

Substitute \( t = 3 \) into the function. This results in the equation: \( D(3) = 50 e^{-0.2 imes 3} \).

Step 3: Calculate the Exponential Value

Calculate \( -0.2 imes 3 = -0.6 \). This transforms the equation into \( D(3) = 50 e^{-0.6} \).

Step 4: Use Exponential Function

Calculate \( e^{-0.6} \). This can be done using a calculator to find it approximately equal to 0.5488.

Step 5: Final Multiplication

Multiply the result of the exponential function by 50:\[ D(3) = 50 imes 0.5488 \approx 27.44 \] milligrams.

Key Concepts

Exponential Function

An exponential function is a mathematical expression involving a constant base raised to a variable exponent. In the context of drug dosage, the continuous decay of the medicine in a patient's bloodstream can be represented using such a function. For the given problem, the function used is \( D(t)=50e^{-0.2t} \). Here, \( e \) (approximately 2.718) is known as Euler's number. It is the base of the natural logarithm and a fundamental constant in mathematics.

Key characteristics of exponential functions include:

• Rapid Change: The quantity decreases or increases rapidly depending on the exponent.
• Continuous Process: The change happens continuously, without breaks.

Understanding these characteristics helps in modeling real-world processes like drug elimination.

Modeling with Functions

Mathematical functions serve as powerful tools to model real-world scenarios. They allow for the prediction and analysis of complex systems by using simple equations. In this exercise, the function \( D(t) = 50e^{-0.2t} \) models how a drug's concentration decreases over time in a patient's bloodstream.

Benefits of using functions for modeling include:

• Simplicity: Complex processes can be simplified into understandable equations.
• Predictability: They enable predictions about future behavior or conditions.
• Analysis: Functions help in analyzing rates of change, maxima, minima, and inflection points.

With such strengths, functions support various fields beyond biology, like engineering and finance.

Drug Dosage Calculation

Calculating the correct dosage and understanding how it changes over time is crucial in medical applications. In this case, the calculation involves evaluating the function \( D(t) = 50e^{-0.2t} \) for a certain time, offering a way to determine the amount of drug at any given moment.

Steps for drug dosage calculation using exponential functions:

• Substitute the time \( t \) into the function.
• Compute the value of the exponent.
• Calculate the exponential expression \( e^{-0.6} \).
• Multiply by the initial dosage to find the remaining amount.

Accurate calculations ensure that patients receive the correct dosage for effective treatment.

Time-dependent Decay

Time-dependent decay refers to the process where the quantity of a substance reduces over time. In pharmacokinetics, it explains how drugs are metabolized and eliminated from the body. The function \( D(t) = 50e^{-0.2t} \) reveals how time affects the drug's concentration.

Features of time-dependent decay include:

• Exponential Nature: Decay often follows an exponential pattern, meaning it decreases rapidly initially and slows down over time.
• Predictability: Knowing the decay rate (given by the coefficient of \( t \) in the exponent) aids in predicting the drug's concentration at any time.
• Application: Beyond pharmacology, time-dependent decay is applicable in areas like radioactive decay and population dynamics.

These insights guide medical professionals in understanding dosage effectiveness and timing.