Problem 57 - Calculus: Early Transcendental Functions

Question

The antibiotic ertapenem has a half-life of 4 hours in the human bloodstream. The dosage is 1 gm per day. Find and graph the amount in the bloodstream \(t\) hours after taking it \((0 \leq t \leq 24)\)

Step-by-Step Solution

Answer

The amount of ertapenem in the bloodstream after \(t\) hours is modelled by the exponential decay function \(A(t) = 1 \cdot e^{-kt},\) with \(k=\frac{\ln(0.5)}{-4}\). The graph of the function is a curve starting at (0,1) and decaying exponentially towards the x-axis over 24 hours.

Step 1: Understanding the half-life concept and starting condition

Given that the half-life of ertapenem is 4 hours, this means that after 4 hours, only half of it remains in the bloodstream. Since the initial amount is 1gm, after 4 hours there will be 0.5gm left. Thus at \(t = 0\), \(A(0) = 1\). This is a condition from which we can construct the decay function.

Step 2: Constructing the decay function

The general form of an exponential decay function is \(A(t) = A_0 \cdot e^{-kt}\), where \(A_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time. Now, we have the initial amount \(A_0 = 1gm\) and decay time \(t = 4\) hours. At \(t = 4\), \(A(4) = 0.5\). Hence, we can substitute these values in the decay function, yielding \(0.5 = 1 \cdot e^{-4k}\). Solving this for \(k\) gives us \(k = \frac{\ln(0.5)}{-4}\).

Step 3: Final decay function

We can now write the decay function as \(A(t) = 1 \cdot e^{-kt},\) where \(k = \frac{\ln(0.5)}{-4}\). This function will give us the amount of antibiotic left in the bloodstream after any number of hours.

Step 4: Graphing the decay function

To graph the function, plot the time \(t\) on the x-axis ranging from 0 to 24 hours (as this represents one day) and the amount \(A(t)\) on the y-axis. You will see that the graph starts at (0,1) and decays exponentially towards the x-axis without ever reaching it. This graph represents how the amount of antibiotic in the bloodstream changes over time.

Key Concepts

Half-Life

In the context of ercn hurts, under utrolified the hubing-ray half-life is a critical concept that describes how quickly the drug decreases in quantity over time. It specifically refers to the time required for the amount of the drug in the bloodstream to reduce by half. For example, if you start with 1 gram of ertapenem, in 4 hours—the half-life—only 0.5 grams will remain. This property allows you to predict how much of the drug is left after a certain period.
Understanding half-life helps in determining the correct dosage of medications and ensures it remains effective without causing harm. It's important in pharmacology as it influences how often you need to administer a dose. In the case of ertapenem, knowing it has a half-life of 4 hours helps healthcare professionals decide on a safe and effective dosing schedule.

Exponential Functions

Exponential functions are mathematical expressions that model how quantities change rapidly over time. In the given problem, we use an exponential decay function to describe the decrease in the amount of ertapenem over time. The general formula for an exponential decay function is:

• \( A(t) = A_0 \cdot e^{-kt} \)
• \( A_0 \) represents the initial quantity of the substance—in this case, 1 gram of ertapenem.
• \( k \) is the decay constant, determining how fast the substance decreases.

When plugging in specific numbers from the problem, like the known half-life, we solve for \( k \) to complete the exponential function. This allows us to calculate precisely how much ertapenem remains in the bloodstream after any given period. Exponential functions are ubiquitous in science and engineering, modeling anything from radioactive decay to population dynamics.

Graphing Techniques

When graphing an exponential decay function like the one describing ertapenem’s concentration over time, it's important to understand how to visually interpret the data. The graph will typically show time (hours) on the x-axis and the amount of drug (grams) on the y-axis. As time progresses, the curve will display a downward trend, starting from the initial amount of 1 gram.
You should expect the graph to follow a smooth curve that gets closer to the horizontal axis (x-axis) but never actually touches it. This represents the concept that the amount of ertapenem will gradually move towards zero, but theoretically, it never entirely vanishes. For visual clarity, points such as the half-life mark (where half the initial amount remains) are helpful to plot, anchoring the decay process visually.
Using graphing techniques effectively helps students and professionals alike easily interpret how changes occur over time and verify calculations or predictions made by the exponential function.