Problem 91
Question
A drug is injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug's concentration, \(y,\) in milligrams per liter, after \(x\) hours is modeled by $$y=\frac{5 x}{x^{2}+1}$$ The graph of this equation, obtained with a graphing utility, is shown in the figure in a \([0,10,1]\) by \([0,3,1]\) viewing rectangle. Use this information. Use the equation to find the drug's concentration after 3 hours. Then identify the point on the equation's graph that conveys this information.
Step-by-Step Solution
Verified Answer
The drug's concentration after 3 hours is 1.5 milligrams per liter. The point on the equation's graph that conveys this information is (3, 1.5).
1Step 1: Substitute the time into the function
To calculate the drug's concentration after 3 hours, substitute \(x = 3\) into the function: \(y = \frac{5 \cdot 3}{3^{2}+1}\).
2Step 2: Simplify the calculation
Simplify the calculation: \(y = \frac{15}{9 + 1}\), which simplifies to \(y = \frac{15}{10}\). So, the concentration of the drug after 3 hours \(y = 1.5\) milligrams per liter.
3Step 3: Identify the point on the graph
Having calculated the value of \(y\) when \(x = 3\), we can represent this information as a point on the graph of the equation. The point is (3, 1.5), where 3 is the number of hours and 1.5 is the drug concentration in the bloodstream in milligrams per liter.
Key Concepts
PharmacokineticsAlgebraic FunctionsRational FunctionsGraphing Utility
Pharmacokinetics
Pharmacokinetics is the science that quantifies the movement of drugs within the body. It considers how a substance is absorbed, distributed, metabolized, and eventually excreted. Key pharmacokinetic processes, often referred to by the acronym ADME, include Absorption, Distribution, Metabolism, and Excretion. Mathematical modeling in pharmacokinetics is essential, as it helps predict the drug concentration in the bloodstream over time based on these processes.
In the given exercise, the formula \(y = \frac{5x}{x^{2}+1}\) models the drug concentration over time. This is a practical application of pharmacokinetic principles. By understanding this equation, medical professionals can determine the most effective dosage and timing for medication administration to optimize therapeutic effects while minimizing potential side effects.
In the given exercise, the formula \(y = \frac{5x}{x^{2}+1}\) models the drug concentration over time. This is a practical application of pharmacokinetic principles. By understanding this equation, medical professionals can determine the most effective dosage and timing for medication administration to optimize therapeutic effects while minimizing potential side effects.
Algebraic Functions
Algebraic functions are mathematical expressions that relate one quantity with another. They consist of constants, variables, and algebraic operations—including addition, subtraction, multiplication, division, and exponents. In the context of pharmacokinetics, algebraic functions help in defining the relationship between drug concentration and time post-administration.
The drug concentration model \(y = \frac{5x}{x^{2}+1}\) is an example of an algebraic function, where \(y\) represents the drug concentration and \(x\) the time elapsed. By substituting different values of \(x\), corresponding values of \(y\) can be determined, thus allowing health practitioners to estimate the drug concentration at any given time.
The drug concentration model \(y = \frac{5x}{x^{2}+1}\) is an example of an algebraic function, where \(y\) represents the drug concentration and \(x\) the time elapsed. By substituting different values of \(x\), corresponding values of \(y\) can be determined, thus allowing health practitioners to estimate the drug concentration at any given time.
Rational Functions
Rational functions are a type of algebraic function characterized by the presence of one polynomial divided by another. The example \(y = \frac{5x}{x^{2}+1}\) from the exercise is a rational function because the numerator (5x) and the denominator (\(x^{2}+1\)) are both polynomials.
Rational functions often model processes that steadily increase or decrease and then level off, as seen in many biological and chemical reactions. This leveling off, or asymptotic behavior, represents a state where changes start to have less impact on the function's value. For a drug in the body, this might represent a point beyond which increasing the time doesn't significantly increase the concentration due to the body's metabolic limits.
Rational functions often model processes that steadily increase or decrease and then level off, as seen in many biological and chemical reactions. This leveling off, or asymptotic behavior, represents a state where changes start to have less impact on the function's value. For a drug in the body, this might represent a point beyond which increasing the time doesn't significantly increase the concentration due to the body's metabolic limits.
Graphing Utility
A graphing utility is a tool that allows us to visualize algebraic and rational functions, making it easier to understand their behavior. By plotting the function \(y = \frac{5x}{x^{2}+1}\) in a graphing utility, we obtain a graphical representation of how drug concentration changes over time.
The graph helps users quickly identify key features such as maximum concentrations, the time it takes to reach that maximum, and how concentration diminishes. It also enables the immediate location of specific values, such as finding the concentration after 3 hours by identifying the point (3, 1.5), where '3' is the elapsed time and '1.5' milligrams per liter represents the corresponding drug concentration.
The graph helps users quickly identify key features such as maximum concentrations, the time it takes to reach that maximum, and how concentration diminishes. It also enables the immediate location of specific values, such as finding the concentration after 3 hours by identifying the point (3, 1.5), where '3' is the elapsed time and '1.5' milligrams per liter represents the corresponding drug concentration.
Other exercises in this chapter
Problem 90
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