Problem 83 - Algebra A Combined Function

Question

The dose of medicine prescribed for a child depends on the child's age \(A\) in years and the adult dose \(D\) for the medication. Young's Rule is a formula used by pediatricians that gives a child's dose \(C\) as $$ C=\frac{D A}{A+12} $$ Suppose that an 8-year-old child needs medication, and the normal adult dose is \(1000 \mathrm{mg}\). What size dose should the child receive?

Step-by-Step Solution

Answer

The child should receive a dose of 400 mg.

Step 1: Identify the given values

The problem provides the following values: the child's age \( A = 8 \) years and the adult dose \( D = 1000 \) mg.

Step 2: Understand Young's Rule Formula

The formula given is: \[ C=\frac{D \cdot A}{A+12} \]where \( C \) is the child's dose. We will use this formula to calculate the child's dose.

Step 3: Substitute the values into the formula

Insert the given values into the formula:\[ C = \frac{1000 \times 8}{8 + 12} \]

Step 4: Calculate the denominator

Compute the value of the denominator:\[ A + 12 = 8 + 12 = 20 \]

Step 5: Calculate the child's dose

Using the formula, calculate the child's dose:\[ C = \frac{1000 \times 8}{20} = \frac{8000}{20} = 400 \]Therefore, the child should receive a dose of \( 400 \) mg.

Key Concepts

Young's Rule

When it comes to adjusting adult medication dosages for children, precision is key. This is where Young's Rule comes into play. By using this formula, healthcare professionals can calculate a child's medication dose based on their age and the standard adult dosage. This ensures the child's dose is both safe and effective.

Young's Rule is expressed through the equation:

• \( C = \frac{D \cdot A}{A + 12} \)

Where:

• \( C \) is the child’s dose
• \( D \) is the adult dose
• \( A \) is the child’s age in years

Using Young's Rule:
To better understand how this works, let's use the example given in the exercise: For a child of 8 years, with an adult dose set at 1000 mg. By substituting these values into the formula, the calculation becomes simplified and straightforward:
\[C = \frac{1000 \times 8}{8 + 12}\]
As shown, this approach not only guarantees accuracy but also prevents potential overdose risks for the young ones. It's a reliable method widely used in pediatric care.

fractions in algebra

Fractions are a common element in algebra, often used to represent parts of a whole or ratios. When dealing with algebraic fractions, understanding how to manipulate and simplify them is crucial for solving equations efficiently.

Basic Concepts:

• A fraction consists of a numerator (top number) and a denominator (bottom number).
• For example, in the fraction \( \frac{1000 \times 8}{20} \), \(1000 \times 8\) is the numerator and \(20\) is the denominator.
• Simplifying a fraction involves finding an equivalent fraction in its simplest form by dividing both the numerator and the denominator by their greatest common factor.

In the context of Young's Rule, fractions help us calculate the proper dosage by balancing parts. This involves performing operations such as multiplication and division within the fractional setup.

Working through the Exercise:
Suppose we computed the expression \( \frac{8000}{20} \). By simplifying this, we find:

• Division leads to each part being reduced to \( 400 \), which is the resulting dosage.

Understanding fractions aids in various applications, from finance to medicine, proving its utility in many everyday calculations.

substitution method in algebra

The substitution method is a fundamental technique in algebra, used primarily to solve systems of equations or evaluate expressions by replacing variables with their corresponding values. This process enables us to simplify equations and reach solutions effectively.

Understanding Substitution:

• Start by identifying the variable to substitute – in this context, parameters like the child's age and the adult dosage.
• Replace the variable with known values to simplify the expression.

In our exercise, substitution is key in applying Young's Rule. Let's break down the steps:

• First, we label our known quantities: \( A = 8 \) and \( D = 1000 \).
• Substitute these directly into the formula, \( C = \frac{D \cdot A}{A + 12} \).
• This leads to a straightforward calculation: \( C = \frac{1000 \times 8}{8 + 12} \).

By performing substitution, complex equations are converted into manageable arithmetic, allowing for an easier path to finding the child's correct dosage. This method is essential in various mathematical scenarios, ensuring clarity and precision.