Problem 64 - Introductory Algebra for College Students

Question

We considered two formulas that approximate the dosage of a drug prescribed for children. $$\begin{aligned}&\text { Young's rule: } C=\frac{D A}{A+12}\\\&\text { Cowling's rule: } C=\frac{D(A+1)}{24}\end{aligned}$$.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. At which age, to the nearest tenth of a year, do the two formulas give the same dosage?

Step-by-Step Solution

Answer

The two formulas give the same dosage when the child is 12 years old.

Step 1: Setup the equation

Firstly, you should set up the equation; The problem tells us that at a certain age both Young’s rule and Cowling’s rule give the same dosage, which means \( \frac{D A}{A+12} = \frac{D(A+1)}{24} \). Here, A represents the child's age, D is a common factor and can be cancelled out leading to \( A / (A+12) = (A+1) / 24 \).

Step 2: Clear the denominator

To make the equation easier to solve, clear out the denominator in the equation by multiplying both sides by \((A+12)*24\) leading to \( 24A = (A+12)*(A+1) \).

Step 3: Solve the equation

Continue to simplify the equation to its simplest form \( 24A = A^2 + 13A + 12 \) and then we state it as a quadratic equation by moving all terms to one side of the equation which gives \( A^2 - 11A + 12 = 0 \).

Step 4: Solve for A

To solve the quadratic equation, find the values for A that will let the equation hold. Factoring the equation gives \( (A-1)(A-12) = 0 \). Hence, the solutions are A = 1 and A = 12.

Step 5: Check for the appropriate solution

The problem tells us that the formulas apply for ages 2 through 13 inclusive, hence the appropriate value within this range is A = 12.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in various mathematical contexts. They are equations where the highest exponent of the variable is 2, typically written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving these equations involves finding the values of \( x \) that make the equation true. These values are known as the roots of the equation.
In the context of the exercise, we have the quadratic equation \( A^2 - 11A + 12 = 0 \) derived from the Young's and Cowling's dosage rules. Quadratic equations like this one can usually be solved by factoring, using the quadratic formula, completing the square, or graphically plotting the function.
For this specific problem, factoring was the chosen method, breaking down the equation into \((A-1)(A-12) = 0\). The solutions to this are the ages that equate the two dosage rules, but only ages within 2 to 13 are valid, giving us age 12 as the final answer.

Dosage Calculation

Dosage calculation is essential in ensuring safe and effective medication administration, especially for vulnerable populations like children. It involves determining the correct amount of medication based on various factors, including age, weight, and the recommended adult dosage. Without precise calculations, there is a risk of underdosing or overdosing.
In this exercise, dosage calculations are performed to equate two distinct formulas, Young's and Cowling's rules, which aim to estimate the appropriate dosage for children. Each formula takes into account the adult dosage and adjusts it based on the child's age.

• Young’s Rule formulates the dosage as \( C = \frac{D \cdot A}{A+12} \)
• Cowling’s Rule formulates the dosage as \( C = \frac{D(A+1)}{24} \)

These formulas help to ensure that children between the ages of 2 and 13 receive an appropriately reduced dose suited to their age.

Young's Rule

Young's Rule is applied in pediatric medication dosing to adjust the adult dosage for a child's age. The rule provides a simpler estimate by considering only the child's age in years, not requiring additional parameters like weight.
Young’s Rule is expressed by the formula \( C = \frac{D \cdot A}{A + 12} \), where:

• \( C \) is the child's dosage.
• \( D \) is the adult dosage.
• \( A \) is the child's age in years.

This formula implies that as the child's age increases, the dosage approaches the adult dosage, making it a practical approach for calculating doses when weight is unknown or irrelevant. It assumes that a child's capacity to process medication increases in a manner proportionate to their age.

Cowling's Rule

Cowling's Rule is another pediatric dosing strategy that modifies the adult dosage considering a child's age. Like Young's Rule, it provides a straightforward method to calculate safe and suitable drug doses for children, primarily adjusted for age.
Cowling's Rule uses this formula: \( C = \frac{D(A+1)}{24} \), where:

• \( C \) is the calculated child's dosage.
• \( D \) is the adult dosage.
• \( A \) is the child's age in years.

This formula simplifies the dosing process, offering a slightly different approach than Young's Rule by adding 1 to the child's age and dividing by a constant 24. It is particularly useful in clinical settings where quick and simple calculations are necessary for effective pediatric care. Keep in mind, however, these are estimations, and professional discretion should be considered for each patient's unique needs.