Problem 102 - College Algebra with Modeling and Visualization

Question

The concentration of a drug in a medical patient's bloodstream is given by the formula \(f(t)=\frac{5}{t^{2}+1},\) where the input \(t\) is in hours, \(t \geq 0,\) and the output is in milligrams per liter. (a) Does the concentration of the drug increase or decrease? Explain. (b) The patient should not take a second dose until the concentration is below 1.5 milligrams per liter. How long should the patient wait before taking a second dose?

Step-by-Step Solution

Answer

(a) The drug concentration decreases over time. (b) Wait over 1.53 hours for a second dose.

Step 1: Understanding the Function

The concentration of the drug is represented by the function \( f(t) = \frac{5}{t^2 + 1} \), where \( t \) is time in hours. As \( t \) increases, we are interested in whether the concentration increases or decreases.

Step 2: Analyze the Behavior of the Function

Examine the function \( f(t) = \frac{5}{t^2 + 1} \). As \( t \) increases, the denominator \( t^2 + 1 \) increases, causing \( \frac{5}{t^2 + 1} \) to decrease. Thus, the concentration of the drug decreases over time.

Step 3: Set the Concentration Threshold Equation

To find out when the concentration falls below 1.5 mg/L, we set the inequality \( \frac{5}{t^2 + 1} < 1.5 \). This inequality helps us find the needed time \( t \).

Step 4: Solve the Inequality

Multiply both sides of the inequality \( \frac{5}{t^2 + 1} < 1.5 \) by \( t^2 + 1 \) to get \( 5 < 1.5(t^2 + 1) \). Simplify this to \( 5 < 1.5t^2 + 1.5 \). Next, subtract 1.5 from both sides to obtain \( 3.5 < 1.5t^2 \).

Step 5: Solve for t

Divide both sides of \( 3.5 < 1.5t^2 \) by 1.5: \( \frac{3.5}{1.5} < t^2 \), which simplifies to \( 2.33 \approx 2.33 < t^2 \). Finally, take the square root of both sides to find \( t > \sqrt{2.33} \approx 1.53 \).

Step 6: Conclusion on Wait Time

The patient should wait for more than \( 1.53 \) hours to ensure the drug concentration falls below 1.5 mg/L before taking a second dose.

Key Concepts

Inequalities

In College Algebra, inequalities are a fundamental concept. They allow us to solve problems where certain conditions must be met, such as in the exercise with drug concentrations. To understand this, let's break down what an inequality is.
Inequalities are similar to equations but use signs like '<', '>', '≤', or '≥'. For instance, solving the inequality \( \frac{5}{t^2 + 1} < 1.5 \) helped us find when the drug concentration becomes less than 1.5 mg/L. The inequality tells us not just if two expressions are equal, like an equation, but whether one expression is larger or smaller than another.
Here's how the process works:

• We first multiply both sides of the inequality by \( t^2 + 1 \) to eliminate the fraction. This is because multiplying by a positive term doesn't change the direction of the inequality.
• Next, we simplify and isolate \( t^2 \) on one side.
• Finally, we solve for \( t \) to determine when the inequality holds true.

Understanding inequalities is crucial as they are used in setting limits and thresholds in real-world applications, such as determining safe drug dosage times.

Functions

Functions represent relationships between two variables, generally written as \( f(x) \). They are a core aspect of College Algebra, describing how one quantity changes with another. In this exercise, the function \( f(t) = \frac{5}{t^2 + 1} \) models the drug concentration over time.
A function assigns every input exactly one output. When studying this function, note these key points:

• The numerator '5' is constant, meaning the concentration is inversely related to \( t^2 + 1 \).
• As time \( t \) increases, the denominator \( t^2 + 1 \) also increases, causing the concentration to decrease.
• This decreasing function aligns with a logical outcome: over time, the drug becomes diluted in the bloodstream.

By understanding the behavior of functions, you can predict trends and changes, which is essential in making decisions in medicine and other fields.

Drug Concentration Model

The drug concentration model described in this exercise uses a function to simulate how a drug disperses within a patient's body over time. This model is essential for ensuring safe and effective medication usage.
Here's how to interpret this model:

• At \( t = 0 \), the concentration is at its maximum \( \frac{5}{1} = 5 \) mg/L, indicating the initial dosage.
• With increasing time \( t \), \( t^2 + 1 \) grows, resulting in a decreased concentration \( \frac{5}{t^2 + 1} \).
• Understanding when the concentration drops below a critical threshold, like 1.5 mg/L, helps decide when it is safe to administer another dose.

This drug concentration model is not only applicable in real-world medical scenarios but also illustrates how algebraic models aid in planning and decision-making. It showcases how mathematical concepts are critical in health sciences for creating effective treatment plans.