Problem 22 - Calculus Early Transcendentals

Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. A drug is eliminated from the body at a rate of \(15 \% / \mathrm{hr} .\) After how many hours does the amount of drug reach \(10 \%\) of the initial dose?

Step-by-Step Solution

Answer

Answer: It takes approximately 14.12 hours for the drug to decrease to 10% of its initial dose with a 15% decay rate per hour.

Step 1: Write the general equation for exponential decay

First, let's write the general equation for exponential decay: \(A(t) = A_0 * (1 - r)^t\) where \(A(t)\) is the amount of drug at time \(t\), \(A_0\) is the initial amount of drug or the reference point (in our case, at \(t=0\)), \(r\) is the decay rate (in this scenario, 15% per hour), and \(t\) is the time in hours.

Step 2: Substitute the decay rate

Now, we substitute the decay rate of 15% into the equation: \(A(t) = A_0 * (1 - 0.15)^t\) The decay rate of 15% is equivalent to 0.15 as a decimal.

Step 3: Solve for the time when the drug's amount reaches 10% of the initial dose

We want to find the time \(t\) when the amount of the drug reaches 10% of the initial dose (\(A_0\)). So, we can set \(A(t)\) to \(0.10A_0\) and solve for \(t\): \(0.10A_0 = A_0 * (1 - 0.15)^t\) Divide both sides by \(A_0\): \(0.10 = (1 - 0.15)^t\) Now, we take the natural logarithm of both sides: \(\ln(0.10) = \ln((1 - 0.15)^t)\) Using the property of logarithms, we can bring the \(t\) down: \(t\ln(1 - 0.15) = \ln(0.10)\) Now, isolate \(t\) by dividing by \(\ln(1 - 0.15)\): \(t = \frac{\ln(0.10)}{\ln(1 - 0.15)}\) Calculate the value of \(t\): \(t \approx 14.12\) So, after approximately 14.12 hours, the amount of drug reaches 10% of the initial dose.

Key Concepts

Drug Elimination

Drug elimination refers to the process by which a drug is removed from the body. This is an important concept in pharmacology because it determines how long a drug will remain active in the system. The body eliminates drugs through various mechanisms, primarily metabolism and excretion.
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Elimination rates can vary based on several factors including:

• The chemical properties of the drug
• Individual metabolic rates
• The presence of other substances

A common model for understanding drug elimination is exponential decay, where the drug's concentration decreases at a constant rate over time. This provides a predictable model to determine when a medication will fall below therapeutic levels or be almost completely eliminated from the body. In many cases, drugs are eliminated at a specific percentage per hour, as illustrated by our example of a 15% per hour elimination rate.

Natural Logarithm

The natural logarithm, represented as \(\ln\), is a function that is the inverse of exponential functions and is particularly useful in solving exponential decay problems. Logarithms allow us to work backwards from an output, such as a percentage of a drug left in the body, to find the input, such as time.
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In the case of drug elimination modeled by exponential decay, the natural logarithm helps to solve for the time \(t\). When you have an equation where the unknown \(t\) is an exponent, as in \(A(t) = A_0 \cdot (1 - r)^t\), taking the natural logarithm of both sides can simplify the equation. This results in:\[\ln(0.10) = t \cdot \ln(1 - 0.15)\]Allowing you to isolate and solve for \(t\) using:\[t = \frac{\ln(0.10)}{\ln(1 - 0.15)}\]Natural logarithms play a crucial role in understanding and calculating the duration over which drugs are eliminated from the body. They transform exponential relationships into linear ones, making them easier to analyze and solve.

Decay Rate

The decay rate is a key parameter in modeling exponential decline, such as the elimination of a drug from the body. It is expressed as a percentage that describes the rate at which the drug's concentration decreases over time. For instance, a decay rate of 15% per hour means that every hour, 15% of the remaining drug is metabolized or excreted.
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Understanding the decay rate allows us to predict how quickly the drug concentration decreases, which is vital for identifying dosing schedules and ensuring effectiveness without toxicity. In exponential decay, the decay rate is used in the equation:\[A(t) = A_0 \cdot (1 - r)^t\]Where:

• \(A(t)\) is the amount remaining at time \(t\)
• \(A_0\) is the initial amount
• \(r\) is the decay rate (0.15 for 15%)

This exponential model provides insights into how the drug concentration diminishes, guiding decisions about dosing intervals and therapeutic monitoring. A well-understood decay rate can ensure that medication remains effective while minimizing potential side effects.