Problem 29 - College Algebra

Question

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 find 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

Approximately 43 milligrams remain after 3 hours.

Step 1: Define the Exponential Model Equation

The formula for exponential decay can be expressed as \( A(t) = A_0 \times (1 - r)^t \), where \( A_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is the time in hours. In this problem, \( A_0 = 125 \) milligrams and the decay rate \( r = 0.30 \). So, our equation becomes \( A(t) = 125 \times (1 - 0.30)^t \).

Step 2: Simplify the Decay Factor

Substitute the decay rate into the equation: \( A(t) = 125 \times (0.70)^t \), because \( 1 - 0.30 = 0.70 \). This represents the amount of drug remaining after \( t \) hours.

Step 3: Calculate the Drug Amount After 3 Hours

Substitute \( t = 3 \) into our simplified equation: \( A(3) = 125 \times (0.70)^3 \).

Step 4: Perform the Calculation

Calculate \( (0.70)^3 = 0.343 \). Therefore, \( A(3) = 125 \times 0.343 = 42.875 \).

Step 5: Round to the Nearest Milligram

Round \( 42.875 \) to the nearest whole number. Therefore, the amount of the drug remaining after 3 hours is approximately 43 milligrams.

Key Concepts

Exponential Model

Exponential models are used to describe mathematical situations where variables change at a constant percentage rate over time. They are crucial for numerous real-world applications, including finance, population growth, and the decay of substances like drugs or radioactive materials.

In our scenario, we have a situation involving exponential decay. The doctor prescribed a drug that decreases in the body by a consistent percentage every hour. We'll use the exponential decay formula, which is:

• \( A(t) = A_0 \times (1 - r)^t \)

This equation allows us to calculate the remaining amount of a substance. \( A_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is time. Exponential models are effective because they involve predictable and continuous change patterns.

Decay Rate

Decay rate (

• \( r \)

) represents the percentage of reduction in the substance per unit of time. In exponential decay, this rate is constant, meaning the substance decreases by the same percentage at every time interval.

For our drug decay example, the decay rate is 30%. This means each hour, the amount of drug in the system reduces to 70% of what it was the previous hour. Calculating the decay factor involves subtracting the decay rate from 1, resulting in 0.70 in this case. This factor becomes the base in our exponential function.

Understanding decay rates is vital in contexts like pharmacology, where controlling the concentration of a drug in a patient's system is essential for efficacy and safety.

Exponential Function

The exponential function described here helps us model the decay of the drug over time. By using the equation:

• \( A(t) = 125 \times (0.70)^t \)

we can estimate the quantity of the drug left at any given time \( t \) hours after administration.

This function incorporates the base, which in our case is the decay factor 0.70, reflecting the consistent decay per hour. The power to which the base is raised is the time variable \( t \), which allows the model to account for the time-dependent nature of exponential decay.

Exponentiation in this function captures the compound reduction effect comprehensively, making it an indispensable tool in applied mathematics scenarios involving exponential changes.

Applied Mathematics

Applied mathematics involves using mathematical methods and models to solve practical problems. This can cover countless scenarios from engineering to economics, and one significant application is modeling physical phenomena like decay processes.

In our example of drug decay, applied mathematics becomes a bridge that translates medical data into actionable insights. Using exponential decay models, doctors can predict how much of a drug remains in a patient's system at different times.

This helps in making informed decisions about dosing schedules or assessing when another dose should be administered to maintain therapeutic levels. The use of mathematical models here exemplifies how abstract concepts can lead to tangible advancements in the real world.