Problem 29 - College Algebra

Question

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about \(30 \%\) each hour. Write an exponential model representing the amount of the drug remaining in the patient's system after \(t\) hours. Then use the formula to fi \(\mathrm{d}\) the amount of the drug that would remain in the patient's system after 3 hours. Round to the nearest milligram.

Step-by-Step Solution

Answer

43 mg of the drug remains after 3 hours.

Step 1: Understand the problem

We have an initial amount of 125 mg of a drug that decays by 30% every hour. We need to write an exponential decay model and then find the amount remaining after 3 hours.

Step 2: Write the Exponential Decay Model

An exponential decay function is typically given by \( A(t) = A_0 \cdot (1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time. Here, \( A_0 = 125 \) mg and \( r = 0.30 \). Therefore, the model is \( A(t) = 125 \cdot (1 - 0.30)^t \).

Step 3: Substitute to find the amount after 3 hours

Substitute \( t = 3 \) into the model: \( A(3) = 125 \cdot (1 - 0.30)^3 \). Calculate \( A(3) = 125 \cdot (0.70)^3 \).

Step 4: Calculate the exponential expression

First calculate \( (0.70)^3 \) which equals approximately 0.343. Therefore, \( A(3) = 125 \cdot 0.343 \).

Step 5: Compute the final amount

Multiply the values: \( 125 \cdot 0.343 = 42.875 \). Round this value to the nearest milligram: 43 mg.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent, typically represented as \( f(t) = a \, b^t \). These functions are renowned for modeling growth or decay processes because they exhibit rapid increase or decrease.In our exercise, the function models the decay of a drug over time, with the base less than 1 because of the decay. It's crucial to understand that exponential functions describe situations where the change is proportional to the amount present. Unlike linear functions, which add or subtract a constant amount, exponential functions multiply by a constant factor, making them suitable for modeling situations like population growth, radioactive decay, and of course, drug dissipation rates.

Decay Rate

The decay rate in exponential functions refers to the percentage by which a quantity decreases over a set period. In this exercise, the decay rate is 30%. It means that every hour, the amount of drug in the patient's system shrinks by 30%.Decay rates are often expressed as decimals in formulas. Thus, a 30% decay rate is represented as \( r = 0.30 \) in the decay model equation. This rate determines how quickly the amount decreases with each passing unit of time.

• A high decay rate indicates a fast reduction of the quantity.
• A lower decay rate shows a slower pace of decay.

Understanding how decay rates work is essential when modeling real-world scenarios, such as discharging capacitors or cooling substances.

Exponential Model

An exponential model is a mathematical representation of an exponential function used to describe real-life situations involving exponential decay or growth. In our drug example, the exponential model is given by:\[ A(t) = A_0 \cdot (1 - r)^t \]Where:

• \( A(t) \) represents the amount of drug left after time \( t \).
• \( A_0 \) is the initial amount, which is 125 mg.
• \( r \) is the decay rate, 0.30 here.
• \( t \) is the time in hours.

This model accurately predicts the remaining drug over time by continuously applying the decay rate to the existing amount, allowing you to gauge at any given time, like after 3 hours the patient will have around 43 mg of the drug left.

Mathematical Modeling

Mathematical modeling involves using mathematical equations and concepts to represent and solve real-world problems. In the context of exponential decay, it's about translating a physical process, like drug elimination from the body, into a mathematical form. The benefits of mathematical modeling include:

• Providing clear visual or numerical understanding of the problem.
• Allowing predictions about future trends or results.
• Enabling calculations that would be difficult through experimentation alone.

In this exercise, the mathematical model consists of the exponential decay equation we've developed. This model helps doctors predict how much medication will remain in the system after a given time, aiding in drug administration and treatment planning. Mathematical models, being precise and adaptable, are valuable tools in fields ranging from healthcare to engineering.