Problem 60 - Introductory Algebra for College Students

Question

We have seen that Young's rule $$C=\frac{D A}{A+12}$$ can be used to approximate the dosage of a drug prescribed for children. In this formula, \(A=\) the child's age, in years, \(D=\)an adult dosage, and \(C=\)the proper child's dosage. Use this formula to solve Exercises. When the adult dosage is 1000 milligrams, a child is given 500 milligrams. What is that child's age?

Step-by-Step Solution

Answer

The child's age is 12 years.

Step 1: Identify Known Values

First, identify and clearly state what values we know from the problem. We know that adult dosage \(D = 1000mg\), child dosage \(C = 500mg\). We are asked to find the child's age, which is \(A\) in the formula.

Step 2: Rewrite the equation

We need the age, \(A\), isolated. Thus, we must rewrite the formula to look like this: \(A = \frac{D \cdot 12}{C} - 12 \). We obtain this by multiplying the whole equation by \(A + 12\), thus getting rid of the denominator in the original equation. Then simplify the equation by subtracting 12.

Step 3: Substitute the known values into the equation

Now, substitute \(C = 500mg\), \(D = 1000mg\) into the equation, so \(A = \frac{1000 \cdot 12}{500} - 12\).

Step 4: Calculate the answer

The result of the calculation will give us the unknown age \(A\). Thus, \(A = 24 - 12\).

Key Concepts

Drug Dosage Calculation

Calculating the correct drug dosage is essential for the safe and effective treatment of patients, especially when it comes to vulnerable groups like children. This task often involves using specific formulas or rules to ensure that the medication is administered in the right amount.
One of the fundamental principles in drug dosage calculation is to adjust the dose based on individual characteristics, such as age, weight, or surface area. This is because these factors directly affect how a drug is absorbed, distributed, metabolized, and eliminated from the body.

• For adults, dosage is often standardized, but for children and other specific groups, adjustments are necessary.
• Safety and efficacy are paramount, so accurate calculations are crucial to avoid overdosage or underdosage.

Dosage calculations often involve simple arithmetic or using formulas such as Young's Rule, which factors in a child's age to derive a suitable dosage. In any calculation, it’s vital to ensure that all units are consistent to avoid errors.

Children's Medication

Children respond to medications differently than adults due to their size and developmental stage, which makes exact dosing critical. It is not enough to simply administer a fraction of an adult dosage.

• Metabolic rates vary; children often metabolize drugs quicker than adults.
• Their organs and systems, like the liver and kidneys, handle drugs differently, affecting elimination and potential side effects.

An essential aspect of pediatric care is ensuring that medication is both effective and safe for a child’s age and size. Young's Rule is one of several methods that can be used to determine the correct dosage for children by incorporating the child's age as a key factor.

Age-Based Dosage Formula

The age-based dosage formula, such as Young's Rule, is a simple and effective tool used to calculate the appropriate dosage of medication for children. Young's Rule is expressed mathematically as:\[ C = \frac{D \cdot A}{A + 12} \] This formula helps determine the child-specific dosage by considering:

• \(C\) - Child's dosage
• \(D\) - Adult's dosage
• \(A\) - Child's age in years

To find a child's age when given a specific dosage, the formula can be rearranged to isolate \(A\):\[ A = \frac{D \cdot 12}{C} - 12 \] Using this approach ensures that medication is administered in a dose appropriate for the child’s age, contributing significantly to the efficacy and safety of the treatment. This simple algebraic manipulation makes Young's Rule a practical tool in pediatrics.