Problem 94 - Algebra and Trigonometry

Question

Use the exponential decay model, \(A=A_{0} e^{b x},\) to solve this exercise. The half-life of aspirin in your bloodstream is 12 hours. How long, to the nearest tenth of an hour, will it take for the aspirin to decay to \(60 \%\) of the original dosage?

Step-by-Step Solution

Answer

It will take approximately 17.3 hours for the aspirin to decay to 60% of the original dosage.

Step 1: Identify the known quantities

The half-life \(T_{1/2} = 12\) hours. Also, the original dosage decays to 60%, so \(A/A_{0} = 0.6\). We want to find the time \(x\).

Step 2: Set up the decay formula using these values

We know that the decay formula is given by \(A=A_{0} e^{bx}\). At half-life, \(A = A_{0}/2\). Substituting these values in the equation, \((1/2) = e^{12b}\). Taking logarithm on both sides, we get \(b = ln(1/2) / 12\).

Step 3: Solve for the time

We substitute the calculated \(b\) value into decay formula to find the time. So, the equation becomes \(0.6 = e^{x * ln(1/2) / 12}\). We solve this for \(x\), by firstly taking the natural logarithm of both sides and then multiplying with the reciprocal of \(b\). This gives \(x = ln(0.6) / b\).

Step 4: Calculate

This results in \(x= 17.3\) after rounding to the nearest tenth of an hour.

Key Concepts

Understanding the Half-Life Formula

The half-life of a substance is a pivotal concept in understanding how substances degrade or decay over time, especially in fields such as chemistry, physics, and pharmacology. The half-life is defined as the time required for a quantity to reduce to half its initial value. This concept is crucial when dealing with radioactive decay or medication in the bloodstream, which is predictable by an exponential decay model.

In our aspirin example, we know the half-life is 12 hours. To compute the time it takes for the drug to decay to a certain percentage of the original amount, we use the half-life formula integrated into the exponential decay model. The relationship can be expressed as follows:
\[ A = A_{0} \times 0.5^{(x/T_{1/2})} \]
where:\( A \) is the amount remaining after time \( x \), \( A_{0} \) is the original amount, \( T_{1/2} \) is the half-life, and \( x \) is the time elapsed. By rearranging the formula and using natural logarithm calculations, we can solve for any of the variables given the others.

Natural Logarithm Calculations

The natural logarithm is the inverse of the exponential function with base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. In many scientific and mathematical computations, particularly those involving growth and decay, the natural logarithm (denoted as \( \text{ln} \)) plays a significant role.

When solving for variables in an exponential decay model, one often takes the natural logarithm of both sides of the equation to simplify the expression. This operation is essential when isolating the exponent in an exponential equation such as \( e^{bx} \). For example, if we have \( e^{bx} = y \), taking the natural log of both sides gives us:\
\[ \text{ln}(e^{bx}) = \text{ln}(y) \]
Using the property that \( \text{ln}(e^{bx}) = bx \), we can then solve for \( x \) given \( y \) and \( b \). This step is crucial in determining the time it takes for something to decay, like the aspirin in our bloodstream.

Exponential Decay in Pharmacokinetics

Pharmacokinetics involves the study of how drugs move within the body, and one of its primary concerns is the metabolism and excretion of substances. The principle of exponential decay is widely used to describe how drugs are cleared from the body. It takes into consideration factors such as the half-life, which can inform doctors and pharmacists how frequently a medication should be administered.

In practical terms, this means understanding how quickly the drug's concentration drops in the body, ensuring the dosage remains therapeutic but not harmful. Applying the exponential decay formula:\
\[ C(t) = C_{0} \times e^{-kt} \]
where \( C(t) \) is the drug concentration at time \( t \), \( C_{0} \) is the initial concentration, and \( k \) is the elimination rate constant, we can predict the drug level at any given time. This model is a simplification but is valuable for understanding the general pattern of how drugs dissipate in the system. In relation to our aspirin example, knowing the half-life and desired percentage of the drug remaining helps in scheduling proper dosage intervals for maintaining its efficacy.