Question

Most drugs in the bloodstream decay according to the equation \(y^{\prime}=c y\), where \(y\) is the concentration of the drug in the bloodstream. If the half-life of a drug is 2 hours, what fraction of the initial dose remains after 6 hours?

Short answer

The fraction of the initial dose remaining after 6 hours is \(\frac{1}{8}\).

Step-by-step solution

Understanding the Drug Decay Equation

The equation given is \( y' = cy \), which is a differential equation describing exponential decay. Here, \( y \) is the concentration of the drug in the bloodstream and \( c \) is a constant that needs to be determined.

Solving the Differential Equation

The solution to the differential equation \( y' = cy \) is \( y = y_0 e^{ct} \), where \( y_0 \) is the initial concentration of the drug.

Using the Half-life to Find c

The half-life of the drug is given as 2 hours, meaning that after 2 hours, the concentration should be half of the initial concentration. Therefore, \( \frac{y_0}{2} = y_0 e^{2c} \).Simplifying, we get:\[ \frac{1}{2} = e^{2c} \]Taking the natural logarithm on both sides, we have: \[ \ln{\frac{1}{2}} = 2c \]Thus, \( c = \frac{\ln{\frac{1}{2}}}{2} = -\frac{\ln{2}}{2} \).

Finding the Fraction After 6 Hours

Now, we want to find the fraction of the drug remaining after 6 hours. Substitute \( c = -\frac{\ln{2}}{2} \) and \( t = 6 \) into the formula \( y = y_0 e^{ct} \). We are looking for \( \frac{y}{y_0} = e^{6c} \).\[ \frac{y}{y_0} = e^{6 \cdot \left(-\frac{\ln{2}}{2}\right)} = e^{-3 \ln{2}} = (e^{-\ln{2}})^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \].

Final Step: Conclusion

Therefore, the fraction of the initial drug dose remaining after 6 hours is \( \frac{1}{8} \), which means 12.5% of the original dose remains in the bloodstream.

Key concepts

Half-life

Half-life is a term used to describe the time required for a quantity to reduce to half its initial value. In the context of drug concentration, it means the concentration of a drug in the bloodstream decreases to half every specified amount of time.

For example, if a drug has a half-life of 2 hours, this means that every 2 hours, the amount of the drug in the bloodstream will be halved. This concept is crucial because it helps determine dosing schedules and ensures that a drug remains effective within the body.

Practically, this means:

• After one half-life, 50% remains.
• After two half-lives, 25% remains.
• After three half-lives, 12.5% remains.

Understanding half-life helps in finding how long a drug remains active in the system or how long it will take to fall below therapeutic levels.

Differential Equation

Differential equations are mathematical equations that describe the rate at which things change. In our exercise, the equation is \( y' = cy \). This represents how the concentration (\( y \)) changes with time.

A differential equation can often be tricky but is useful for modeling real-world processes, like how a drug concentration decreases over time. By solving a differential equation, we can predict future behavior, such as how much drug will remain in the system after a certain time.

The solution to this particular equation is an exponential function, \( y = y_0 e^{ct} \), which we use to find how the drug concentration changes based on its initial amount and the decay rate \( c \).

Exponential Functions

Exponential functions are mathematical expressions used to describe growth or decay, where a quantity increases or decreases rapidly at a rate proportional to its current value. The general form is \( y = a e^{bx} \), where \( a \) is the initial value, and \( b \) controls the rate of change.

In the context of our drug decay problem, the exponential function \( y = y_0 e^{ct} \) is applied. This function describes how the drug concentration diminishes over time.

Exponential decay is characterized by:

• A constant decay factor that results in a rapid decrease initially.
• The rate of change proportional to current value - as the drug concentration lowers, the decay slows down.

Understanding exponential functions is key when solving decay problems as it formalizes how quantities reduce progressively.

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is the inverse of the exponential function. It's a logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. In the context of exponential decay, the natural logarithm helps solve equations involving powers of \( e \).

For example, when working with half-life in drug decay, we use the natural logarithm to isolate \( c \) in the differential equation solution: \( \ln \left( \frac{1}{2} \right) = 2c \).

Important properties include:

• \( \ln(e^x) = x \)
• The natural log changes multiplication into addition (\( \ln(ab) = \ln a + \ln b \))

Understanding natural logarithms is essential when solving for decay constants and interpreting exponential decay models effectively.