Problem 540 - Precalculus

Question

For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17\(\%\) 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 24 hours. Round to the nearest hundredth of a gram.

Step-by-Step Solution

Answer

After 24 hours, approximately 6.76 mg of the drug remains in the patient's system.

Step 1: Understand the Exponential Decay Problem

The scenario presents a situation where the drug amount decreases exponentially over time. Initially, 300 mg of the drug is prescribed, and it decays at a rate of 17\(\%\) per hour, meaning the drug amount decreases by 17\(\%\) each hour. The task involves formulating an exponential model and calculating the amount after a specified period.

Step 2: Define the Exponential Decay Model

The formula to represent exponential decay is given by \( A(t) = A_0 \cdot e^{-kt} \), where \( A(t) \) is the amount remaining at time \( t \), \( A_0 \) is the initial amount, and \( k \) is the decay constant.We start by converting the percentage decay rate into a decay constant: Since the drug decays at a rate of 17\(\%\) per hour, the decay constant \( k \) can be written as 0.17.

Step 3: Formulate the Exponential Model

Given the decay rate of 17\(\%\) per hour, we can write the exponential decay model as: \[ A(t) = 300 \cdot (1 - 0.17)^t \].This equation represents the amount of drug remaining after \( t \) hours, factoring the 17\(\%\) decay per hour.

Step 4: Calculate the Amount After 24 Hours

Substitute \( t = 24 \) into the exponential model to find the amount of the drug remaining after 24 hours: \[ A(24) = 300 \cdot (1 - 0.17)^{24} = 300 \cdot (0.83)^{24} \].Now, calculate the value of this expression.

Step 5: Compute the Final Amount

Evaluate the expression \( (0.83)^{24} \), then multiply by 300 to find the remaining amount: \[ A(24) \approx 300 \cdot 0.02252 \approx 6.7556 \].Round this amount to the nearest hundredth of a gram for the final result.

Key Concepts

Exponential Model

In scenarios involving exponential decay, we often use the exponential model: \[ A(t) = A_0 \cdot e^{-kt}\]\ where:

• \(A(t)\) represents the amount remaining after time \(t\).
• \(A_0\) is the initial amount at time \(t=0\).
• \(k\) is the decay constant, a key parameter that defines the rate of decay over time.

The nature of exponential decay means that the quantity reduces at a rate proportional to its current value. This leads to a rapid decrease initially, which then slows down over time. Knowing how to construct and use exponential models is crucial for problems like calculating drug concentration over time.

Decay Constant

The decay constant \(k\) is fundamental to understanding exponential decay. It simplifies how we represent the percentage rate of decay in mathematical terms. The decay constant is derived from the percentage decay rate using the formula: \[ k = \frac{\text{decay rate}}{100} \] For example, in the given exercise, since the decay rate is 17\(\%\), the decay constant \(k\) becomes 0.17. This value neatly incorporates the percentage, allowing us to use continuous mathematical processes rather than discrete ones. The decay constant provides a concise way to express complex real-world processes.

Percentage Decay Rate

Percentage decay rate is the percentage by which the quantity decreases per unit time. In the drug decay scenario, the rate given is 17\(\%\) per hour.

• The percentage decay rate directly influences the decay constant.
• Converting this rate into a decay constant (by dividing by 100) allows for simpler integration into mathematical models like the exponential decay formula.

An understanding of how the percentage decay rate translates into a decay constant can greatly simplify mathematical computations in problems involving exponential decay.

Formulating Equations

Formulating equations in exponential decay involves creating a mathematical model that accurately represents the scenario. To form an exponential decay equation:

• Identify the initial amount, \(A_0\).
• Determine the decay constant \(k\) from the given percentage decay rate.
• Formulate the equation using the general exponential decay formula \(A(t) = A_0 \cdot (1 - \text{decay rate})^t\).

In the drug example, the initial amount is 300 mg, and the decay rate is 17\(\%\), so the model becomes:\[ A(t) = 300 \cdot (1 - 0.17)^t\] Using such models enables precise computation of values at any point \(t\), facilitating predictions about the system's future behavior.