Problem 50 - Precalculus: Functions and Graphs

Question

Young's rule is a formula that is used to modify adult drug dosage levels for young children. If \(a\) denotes the adult dosage (in milligrams) and if \(t\) is the age of the child (in years), then the child's dose \(y\) is given by the equation \(y=t a /(t+12) .\) Sketch the graph of this equation for \(t>0\) and \(a=100\)

Step-by-Step Solution

Answer

Graph the curve of \( y = \frac{100t}{t + 12} \) which gradually increases and approaches 100 as age \( t \) increases.

Step 1: Substitute Adult Dosage into Equation

Substitute the given adult dosage of \( a = 100 \) milligrams into Young's rule equation. The equation becomes:\[y = \frac{t \times 100}{t + 12}\]

Step 2: Plot the Function

Recognize that the equation \( y = \frac{100t}{t + 12} \) represents a rational function. The graph of this function will be a curve, where the child dosage \( y \) changes with age \( t \).

Step 3: Identify Key Characteristics of the Graph

Notice that when the child is at a very young age (\( t \) close to zero), \( y \) approaches zero because the weight of \( t \) in the numerator is small compared to the constant \( 12 \) in the denominator. As \( t \) increases, \( y \) increases, approaching 100 (the adult dose) as \( t \) grows larger.

Step 4: Draw the Graph

On a Cartesian coordinate system, draw the horizontal and vertical axes. Label the x-axis as \( t \) (age of the child in years) and the y-axis as \( y \) (the child's drug dosage in milligrams). Plot the curve based on the equation, starting with the point near the origin (as \( t \) approximates zero) and rising as \( t \) increases, getting closer to \( y = 100 \) without actually reaching it.

Key Concepts

Rational Functions

Rational functions are expressions involving the division of two polynomials. In the exercise, the given formula \( y = \frac{100t}{t+12} \) is a rational function. This means it involves a fraction where both the numerator and the denominator are polynomials.

• The numerator \( 100t \) is a linear polynomial.
• The denominator \( t + 12 \) is also a linear polynomial.

When working with rational functions, it's crucial to consider the behavior of the function as the variable changes. One characteristic of these functions is that they can have horizontal asymptotes, which they approach but never quite touch. For the given function, the horizontal asymptote is \( y = 100 \), which you'll see clearly when you sketch the graph. This is due to the degree of the polynomials in the numerator and the denominator being the same; thus, the ratio of the leading coefficients determines the horizontal asymptote.

Dosage Calculations

Dosage calculations involve determining the appropriate amount of medication to give, often particularly challenging when adjusting doses for children. Young's Rule provides a straightforward formula.

• It modifies an adult dosage based on a child's age.
• The formula is \( y = \frac{t \, a}{t + 12} \), where \( a \) is the adult dose, and \( t \) is the child's age.

This equation shows how dosage needs increase with a child's age. It ensures that the medication level is appropriate and safe. It's essential in clinical settings where pediatric dosages must be accurate to avoid underdosing or overdosing. By substituting specific values, as done in the exercise, you can directly calculate the suitable dosage for any value of \( t \).

Graphing Equations

Graphing equations helps visualize how changes in one variable affect another. For Young's Rule, the graph exhibits how age influences dosage. Here's a simple approach to graph the equation:

• Start by plotting key points: when \( t = 0 \), \( y = 0 \), and when \( t \) is large, \( y \approx 100 \).
• Draw the curve between these points, keeping in mind it never quite reaches \( y = 100 \).

The graph is asymptotic to \( y = 100 \), meaning it approaches the line without touching it. This visual representation gives an immediate understanding of how quickly the dosage increases with age. The steepness of the graph near the origin shows faster growth in required dosage at younger ages, leveling off as age increases.

Age-Based Dosage

Age-based dosage adjustments are crucial for providing the right medication amount to children. Children are not just smaller adults, so simply scaling adult doses based on weight or age without a formula like Young's Rule could lead to error.This rule considers developmental and physiological differences between children and adults:

• Ensures safety across various ages by adjusting drugs to be more age-appropriate.
• Reduces risk of side effects linked to improper dosing.

As demonstrated in the exercise, substituting different values of \( t \) (age) allows for customized dosage calculation. Understanding these principles can help in interpreting how children's dosages of medications should be approached, emphasizing the rule's practical application in real-world scenarios.