Problem 90 - Elementary and Intermediate Algebra

Question

The concentration of a certain prescription drug in an organ after \(t\) minutes is modeled by the function \(f(t)=0.08\left(1-e^{-0.1 t}\right)\) where \(f(t)\) is the concentration at time \(t .\) Find the concentration of the drug at 30 minutes.

Step-by-Step Solution

Answer

The concentration of the drug at 30 minutes is approximately 0.0760.

Step 1: Identify the function and information given

We are given the function \( f(t) = 0.08(1 - e^{-0.1t}) \) which models the concentration of the drug in an organ over time. We need to find the concentration at \( t = 30 \) minutes.

Step 2: Substitute \( t = 30 \) into the function

Replace \( t \) with 30 in the function: \[ f(30) = 0.08 (1 - e^{-0.1 imes 30}) \]

Step 3: Calculate the exponential part

First calculate the exponent: \[ -0.1 \times 30 = -3 \] Now evaluate the exponential function: \[ e^{-3} \approx 0.0498 \]

Step 4: Simplify the expression

Substitute back the calculated exponential into the function: \[ f(30) = 0.08 (1 - 0.0498) \] Subtract inside the parenthesis: \[ f(30) = 0.08 \times 0.9502 \]

Step 5: Multiply to find the concentration

Calculate the final multiplication: \[ f(30) = 0.076016 \] Thus, the concentration of the drug at 30 minutes is approximately 0.0760.

Key Concepts

Drug Concentration Modeling

In the realm of pharmacokinetics, drug concentration modeling plays a vital role in understanding how medications interact within the body over time. This type of modeling helps in predicting the concentration of a drug in a particular organ or tissue at any given moment. Here, the function given, \[ f(t) = 0.08(1 - e^{-0.1t}) \]represents the concentration of a drug in an organ, where

• \( f(t) \) is the concentration at time \( t \)
• \( 0.08 \) is a constant that scales the concentration to match realistic physiological conditions
• and \( e^{-0.1t} \) models the decay of the concentration over time.

This function provides a way to calculate the exact concentration at any given time \( t \), ensuring safe and efficient dosing schedules, thus crucial in clinical environments.

Function Evaluation

Evaluating functions is a core mathematical skill that allows us to find specific output values given certain input values. In this exercise, the task was to evaluate the function \[ f(t) = 0.08(1 - e^{-0.1t}) \]at \( t = 30 \) minutes. This process involves the following simple steps:

• Substitute the input value, here 30, into the function in place of \( t \)
• Compute the exponential part, \( e^{-0.1 imes 30} \), to find \( e^{-3} \)
• Simplify the function by performing arithmetic operations within the parentheses
• Complete the evaluation by multiplying the remaining terms to find the result.

Applying these steps provides not just a mechanistic answer but deepens your understanding of underlying mathematical operations and concepts.

Exponential Decay

Exponential decay is a fundamental concept that describes processes that decrease rapidly at a rate proportional to their current value. The mathematical representation is often written as \[ e^{-kt} \],where \( k \) is a positive constant indicating the decay rate. In our drug concentration model, \( e^{-0.1t} \)represents how quickly the drug concentration decreases over time. The negative sign in the exponent ensures this decline. For instance:

• A smaller value of \( t \) results in \( e^{-0.1t} \) being closer to 1, indicating slower initial decrease.
• As \( t \) gets larger, \( e^{-0.1t} \) approaches zero, signaling a rapid decline.
• The choice of 0.1 as \( k \) controls the steepness of the decay, balancing how quickly the drug concentration diminishes for safety.

Understanding exponential decay is crucial in settings from pharmacology to environmental sciences, wherever rates of decrease over time play a significant role in decision-making.