Problem 63 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's bloodstream \(t\) hr after injection is given by $$ C(t)=\frac{0.2 t}{t^{2}+1} $$ a. Find the horizontal asymptote of \(C(t)\). b. Interpret your result.

Step-by-Step Solution

Answer

a. The horizontal asymptote of the function \(C(t) = \frac{0.2t}{t^2+1}\) is y = 0. b. This result means that as time goes on, the concentration of the drug in the patient's bloodstream approaches zero, indicating that the drug's effect will decrease over time and eventually become negligible.

Step 1: a. Find the horizontal asymptote of C(t).

The horizontal asymptote can be found by calculating the limit of the function C(t) as t approaches infinity: \[ \lim_{t \to \infty} \frac{0.2t}{t^2 + 1} \] As t becomes very large, the dominant terms of the numerator and the denominator are the ones with the highest power of t, i.e. 0.2t and \(t^2\), respectively. Hence, we can rewrite the limit as \[ \lim_{t \to \infty} \frac{0.2t}{t^2} = \lim_{t \to \infty}0.2\cdot \frac{1}{t} \] Now, as t approaches infinity, the fraction \(\frac{1}{t}\) approaches 0, so the limit becomes: \[ 0.2 \cdot 0 = 0 \] Thus, the horizontal asymptote of C(t) is y = 0.

Step 2: b. Interpret the result.

The horizontal asymptote at y=0 means that as time goes on (i.e., \(t\) approaches infinity), the concentration of the drug in the patient's bloodstream approaches zero. In other words, the drug becomes less and less concentrated in the bloodstream over time, ultimately becoming negligible. This suggests that the drug's effect will decrease over time, and eventually it will not have any significant effect on the patient.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial in calculus. It tells us the value that a function approaches as the input (or variable) gets closer to a certain point. When we say the limit of function C(t) as t approaches infinity, we are interested in what happens to the drug concentration as time increases indefinitely.

In the given exercise, to find the limit of the function \( C(t)=\frac{0.2 t}{t^{2}+1} \), we consider the behavior of \( C(t) \) as \( t \) becomes very large, that is, as \( t \) approaches infinity. In this case, terms with lower powers of \( t \) become negligible in comparison to those with higher powers of \( t \) in both the numerator and the denominator. Hence, the function can be simplified, allowing us to see that the drug concentration will approach zero as time goes to infinity. This concept also applies to real-life scenarios, like drug administration, where we can predict the declining concentration of a drug over time.

Horizontal Asymptotes

Horizontal asymptotes provide us with a visual understanding of the behavior of a function as the input either increases or decreases without bound. They are straight lines that the graph of a function approaches but never actually reaches or crosses. The concept is neatly demonstrated in the solution for finding the horizontal asymptote of \( C(t) \).

To find the horizontal asymptote, we calculate the limit of \( C(t) \) as \( t \) approaches infinity (or negative infinity). As shown in the solution, \( C(t) \) approaches zero, hence the horizontal asymptote is the line \( y=0 \). This means that no matter how much time passes, the drug concentration will not go below zero, representing a boundary condition for the function which relates to the physical reality that concentration levels cannot be negative.

Drug Concentration Modeling

Drug concentration modeling is a critical application of calculus in the field of pharmacokinetics, which helps us understand how a drug disperses throughout the body over time.

The function \( C(t) \) represents the concentration of a drug in the bloodstream after \( t \) hours. By examining the behavior of this function using the limit and the concept of a horizontal asymptote, we can conclude that the concentration decreases over time. In practice, this model helps medical professionals determine appropriate dosing schedules and understand the duration of a drug's effectiveness in the body. The notion that the limit of \( C(t) \) as \( t \) approaches infinity is zero aligns with the expectation that the drug will eventually be metabolized or excreted, leading to an insignificant concentration level.