Problem 401 - Precalculus

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 find the amount of the drug that would remain in the patient's system after 3 hound to the nearest milligram.

Step-by-Step Solution

Answer

43 milligrams

Step 1: Understand the Problem

You need to create an exponential decay model for the amount of the drug left in the body over time. You know the initial amount of the drug and the rate at which it decays.

Step 2: Define the Exponential Model

In an exponential decay situation, the amount of substance remaining at time \(t\) is given by \(A(t) = A_0(1 - r)^t\), where \(A_0\) is the initial amount and \(r\) is the decay rate. For this problem, \(A_0 = 125\) mg and \(r = 0.30\). The model becomes \(A(t) = 125(0.70)^t\).

Step 3: Substitute \(t = 3\) into the Model

To find the amount remaining after 3 hours, substitute \(t = 3\) into the model equation \(A(t) = 125(0.70)^t\), resulting in \(A(3) = 125(0.70)^3\).

Step 4: Calculate the Amount of Drug Remaining

Now calculate \(A(3) = 125(0.70)^3\). This equals approximately 125 * 0.343 = 42.875 milligrams.

Step 5: Round the Result

Finally, round 42.875 to the nearest milligram. So, the amount of drug remaining after 3 hours is approximately 43 milligrams.

Key Concepts

Exponential Model

An exponential model is a mathematical way to represent situations where a quantity reduces over time in a consistent and predictable manner. In this case, we're looking at how a therapeutic drug diminishes in a patient's system. The core idea in the exponential model is that the rate of decay is proportional to the current amount, making it a consistent percentage decrease over each time interval.

The formula for exponential decay is expressed as:

• \( A(t) = A_0(1 - r)^t \)

where:

• \( A(t) \) is the amount remaining after time \( t \)
• \( A_0 \) is the initial amount
• \( r \) is the decay rate
• \( t \) is the time in consistent units (like hours)

In our scenario, \( A_0 \) will be given in milligrams as the initial dose prescribed, and the model helps determine how much of this drug remains in the patient's system over time.

Decay Rate

The decay rate is a crucial factor in exponential decay, representing the constant percentage by which the drug quantity is reduced each hour. In this problem, the decay rate is 30\(\%\). This means that every hour, 30\(\%\) of the remaining drug is eliminated from the patient's body.

To incorporate this into the exponential model, the decay rate must be converted into a decimal. So, 30\(\%\) becomes 0.30. Completing the setup of the formula, we subtract the decay rate from 1:

• \( 1 - 0.30 = 0.70 \)

This factor, 0.70, is used in the equation to multiply the remaining amount at each step, effectively capturing the weekly decrease in drug presence over the hours.

Initial Amount

The initial amount is the starting quantity of the drug present in the system, before any decay begins. In this problem, the doctor has prescribed 125 milligrams of the drug. This value is foundational for setting up the exponential model, as it represents \( A_0 \) in the formula.

The initial amount is crucial because it's the baseline from which all future amounts are calculated. To see the effects of decay over time, we apply the exponential model formula:

• \( A(t) = 125(0.70)^t \)

where 125 mg is the starting point for calculations, determining the drug's presence after specific time intervals.

Substitution Method

The substitution method involves plugging specific values into the equation to find the remaining amount of the drug after a certain period. Once the exponential model is defined, you can substitute a value for \( t \) to find how much of the drug will remain in the body after that time.

For this exercise, to find the drug amount after 3 hours:

• Substitute \( t = 3 \) into \( A(t) = 125(0.70)^t \).
• This results in \( A(3) = 125(0.70)^3 \).
• Doing the calculation: \( A(3) = 125 \times 0.343 = 42.875 \).

Finally, round the result to the nearest whole number, since drug measurements in prescriptions usually come as whole numbers, giving you approximately 43 milligrams remaining after 3 hours.