Problem 88 - Introductory Algebra for College Students

Question

Doctors use the rational expression $$\frac{D A}{A+12}$$ to determine the dosage of a drug prescribed for children. In this expression, \(A=\) the child's age and \(D=\) the adult dosage. Use the expression. If the normal adult dosage of medication is 1000 milligrams, what dosage should a 4-year-old child receive?

Step-by-Step Solution

Answer

A 4-year-old child should receive a dosage of 250 milligrams.

Step 1: Understand the Given Expression

The given expression is \( \frac{D A}{A+12} \). In this expression, \(D\) represents the adult dosage, and \(A\) represents the child's age.

Step 2: Substitute the Value

Substitute \(D = 1000\) milligrams (the adult dosage) and \(A = 4\) (the child's age) into the expression. So, the expression now becomes \( \frac{1000 \times 4}{4+12} \).

Step 3: Perform the Calculation

Perform the calculations in the numerator and the denominator separately. Then divide the numerator by the denominator to get the child's dosage. The calculation is performed as follows: \( \frac{4000}{16} = 250 \).

Key Concepts

Understanding Rational Expressions

A rational expression is much like a fraction, but instead of integers, you'll find variables like in an algebraic expression. It's composed of a numerator and a denominator where polynomial expressions dwell. In drug dosage calculations for children, rational expressions are vital because they allow healthcare practitioners to determine the correct amount of medication based on a direct relationship with the patient's age.

For example, the expression \frac{D \times A}{A+12}\ is a rational expression where \(A\) is the child's age and \(D\) is the adult dosage. It illustrates that the dosage for children is proportional to their age in relation to a sum that includes their age and a constant number, in this case, 12. In a very real sense, understanding and simplifying rational expressions can be the difference between effective treatment and potential harm. It is therefore essential to grasp these concepts firmly to ensure accuracy in such critical applications.

Calculations in Pediatric Dosage

The administration of drugs to children is a delicate aspect of medical care. The dosage often can't be estimated based on adult dosages alone, as children's metabolisms and body systems are significantly different. This is where pediatric dosage calculations come into play, using tailored expressions to safely adjust adult dosages. These calculations often involve a constant that adjusts the dosage according to age, ensuring that the drug is sufficiently effective without being hazardous.

For instance, using our given expression \frac{D \times A}{A+12}\, healthcare professionals can calculate the dosage for any given child by simply adjusting for the child's age (\(A\)) and adult dosage (\(D\)). What's important is the reliant relationship between the patient's age and dosage, emphasizing that as a child grows, they require a progressively larger dose, up to a point where there's no more need for such age-specific adjustments.

The Role of Algebraic Expressions in Calculating Dosages

Algebraic expressions serve as the cornerstone for setting up problems that involve variables, constants, and operations like addition, multiplication, and division. In the context of drug dosage calculations, these expressions become essential tools for translating medical guidelines into actionable numbers.

For example, the expression \frac{D \times A}{A+12}\ involves the use of the multiplication operation and presents a way of establishing a ratio between the adult dosage (\(D\)), the child's age (\(A\)), and the sum \(A+12\). This shows how algebraic principles apply to real-world scenarios, helping to determine what is safe and effective for pediatric patients. It allows doctors to take a general guideline (the adult dosage) and refine it into an individualized, age-appropriate dosage for a child, thereby utilizing the precision and flexibility of algebra in serving patient-specific needs.