Problem 96 - Introductory Algebra for College Students

Question

Two formulas that approximate the dosage of a drug prescribed for children are Young's rule: \(\quad C=\frac{D A}{A+12}\) and Cowling's rule: \(\quad C=\frac{D(A+1)}{24}\) In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. Use Cowling's rule to find the difference in a child's dosage for a 12 -year- old child and a 10 -year-old child. Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the model.

Step-by-Step Solution

Answer

The difference in dosages for a 12 year-old child and a 10 year-old child based on Cowling's rule is \(\frac{D}{12}\). This indicates that for each year increase in age, the dosage increases by one twelfth of the adult dosage.

Step 1: Understand and Apply the Cowling’s Rule

First, apply the Cowling's Rule for a 12 year-old child, which will result in a dosage expression, \(C_{12}=\frac{D(12+1)}{24}\). Then, apply Cowling's Rule for a 10 year-old child. The resulting dosage is \(C_{10}=\frac{D(10+1)}{24}\).

Step 2: Find the Difference in Dosages

The difference in dosage for the 12 year-old child and the 10 year-old child will be \(C_{12}-C_{10} = \frac{D(12+1)}{24} - \frac{D(10+1)}{24}\).

Step 3: Simplify the Expression

To further simplify the above expression, we can combine it into a single fraction as \(\frac{D}{24} * (13 - 11) = \frac{D * 2}{24} = \frac{D}{12}\).

Step 4: Interpretation of the Result

The result of the expression, \(\frac{D}{12}\), indicates that the difference in dosage for the 12 year-old child and the 10 year-old child is one-twelfth of the adult dosage (D). This suggests that as the child's age increases, the dosage also increases proportional to the adult dosage.

Key Concepts

Young's Rule

Young's rule is a guideline used in pediatric pharmacology to estimate the appropriate drug dosage for a child, based on the child's age. It is formulated by the expression: \[\begin{equation}C = \frac{D \times A}{A + 12}\end{equation}\]where:

• A is the age of the child in years,
• D is the standard adult dose,
• C is the calculated child's dosage.

This method accounts for the physiological differences between children and adults due to age. It suggests that as children grow, the dosage scales in proportion to their age in relation to an adult's dosage assumption that the adult is considered 12 years older than the child for the purpose of the rule. While it can be a helpful starting point, it's important to remember that other factors, such as weight and overall health, must also be taken into account when determining the correct dosage for children.

Cowling's Rule

Cowling's rule, similar to Young's rule, is another method for calculating the appropriate drug dosage for children, but it slightly modifies the age adjustment factor. The rule's formula is given by: \[\begin{equation}C = \frac{D \times (A + 1)}{24}\end{equation}\]where:

• A is the age of the child in years,
• D is the standard adult dose,
• C is the calculated child's dosage.

The difference between Cowling's and Young's rule is in the added '1' to the child's age (A + 1) and the divisor '24' instead of 'A + 12'. This rule suggests a slightly higher dose compared to Young's rule for younger children, and these variations underscore the importance of choosing the right formula based on clinical judgment and considering other individual factors for each child. Since Cowling's rule is factored by both age and a constant (24), it is crucial for healthcare providers to understand its application and underlying mathematics to ensure the safe administration of medications to pediatric patients.

Rational Expressions

Rational expressions are fractions that involve polynomials in both the numerator and denominator. They are used in various fields of study, including pharmacology, to express relationships and calculate doses. In the context of calculating a child's drug dosage using Cowling's rule, a rational expression takes the form: \[\begin{equation}C = \frac{D \times (A + 1)}{24}\end{equation}\]The expression above becomes especially useful when we need to find the difference between dosages for children of different ages as a single expression. This simplification allows healthcare professionals to understand dosage adjustments without dealing with complex calculations for each case, making it easier to determine how the dose varies with age. Comprehending rational expressions is fundamental in algebraic modeling and ensures precise calculations in practical applications like medicine.

Algebraic Modeling

Algebraic modeling involves representing real-world situations with algebraic expressions, equations, or functions to solve problems. In pharmaceutical calculations, algebraic models are crucial for determining drug dosages. For example, Cowling's and Young's rules use algebraic expressions to model the relationship between a child's age and the safe drug dosage. In the exercise where the difference in dosages for children of different ages is sought, algebraic modeling is utilized to derive a simplified expression by combining terms and reducing the fraction:\[\begin{equation}C_{12} - C_{10} = \frac{D \times (12 + 1)}{24} - \frac{D \times (10 + 1)}{24}\end{equation}\]Simplifying this using algebraic techniques, we obtain:\[\begin{equation}\frac{D}{12}\end{equation}\]This modeling makes it clear that the difference in drug dosage is directly proportional to the adult dose and inversely proportional to a constant, aiding in a straightforward understanding of how dosages should be adjusted. By comprehending algebraic modeling, those in the healthcare field can apply these concepts to ensure safe and effective treatment for pediatric patients.