Problem 90 - Intermediate Algebra

Question

Medicine. 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 0.076.

Step 1: Understand the Function

The function given is \(f(t)=0.08(1-e^{-0.1 t})\). This function represents the concentration of the drug over time \(t\) in minutes. To find the concentration at a specific time, we substitute the value of \(t\) into this function.

Step 2: Substitute 30 into the Function

Plug in \(t = 30\) into the function \(f(t)\). This gives us \(f(30) = 0.08(1-e^{-0.1 imes 30})\).

Step 3: Simplify the Exponent

Calculate the exponent: \(-0.1 \times 30 = -3\). So the expression becomes \(f(30) = 0.08(1-e^{-3})\).

Step 4: Calculate the Exponential Part

Find \(e^{-3}\) using a calculator. This is approximately \(e^{-3} \approx 0.0498\). Substitute back to get \(f(30) = 0.08(1-0.0498)\).

Step 5: Simplify the Expression

Subtract the exponential part from 1 to get \(1 - 0.0498 = 0.9502\). Then, calculate \(0.08 \times 0.9502 = 0.076016\).

Step 6: Finalize the Answer

Thus, the concentration of the drug at 30 minutes is approximately \(0.076\) (to three decimal places).

Key Concepts

Drug Concentration Modeling

In pharmacokinetics, modeling drug concentration over time helps us understand how a medication disperses in the body. The mathematical expression given in the problem, \(f(t)=0.08\left(1-e^{-0.1 t}\right)\), is such a function. It relates the drug's concentration in an organ to the time elapsed since administration. Here:

• \(f(t)\): Represents the drug concentration at time \(t\).
• \(0.08\): The scaling factor, which could represent the maximum achievable concentration under controlled conditions.
• \(e^{-0.1t}\): An exponential function demonstrating how quickly the concentration builds up or diminishes.

This model is important for determining optimal drug dosage and frequency, ensuring effectiveness while minimizing side effects. By plugging time values into the function, we can predict how concentration levels change within critical intervals.

Exponential Decay

Exponential decay refers to the process where a quantity decreases over time at a rate proportional to its current value. In drug concentration, exponential decay models how the drug diffuses out or gets metabolized by the body.

• The term \(e^{-0.1t}\) signifies this decay. The base \(e\) is a constant approximately equal to 2.71828.
• The exponent \(-0.1t\) indicates the decay rate, with a negative sign showing reduction.

This type of decay is common in real-world scenarios like radioactive decay, cooling of hot objects, and natural resource depletion. Understanding exponential decay helps medical practitioners predict when drug levels become therapeutically ineffective. As the drug concentration drops, doses may need adjustment or a new regimen implemented.

Substitution Method in Algebra

The substitution method in algebra is a technique used to evaluate expressions by replacing variables with specific values. In the given exercise, we use substitution to determine drug concentration at specific points in time.

• Identify the variable and the value to be substituted. Here, \(t = 30\).
• Replace \(t\) in the formula \(f(t)=0.08(1-e^{-0.1 t})\) with 30.
• Simplify the expression step by step to find the value of \(f(30)\).

This method is crucial for solving many algebraic problems and equations, not just drug concentration models. The key is simplifying the equation gradually, ensuring you carefully perform each mathematical operation in order to achieve accurate results. By methodically substituting and calculating, complex problems become more manageable and easier to solve.