Problem 87 - 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 an 8 -year-old child receive?

Step-by-Step Solution

Answer

The dosage for an 8-year-old child should be 400 milligrams.

Step 1: Identify the Known Variables

From the problem, we know the adult dosage \(D\) is 1000 milligrams and the child's age \(A\) is 8.

Step 2: Substitute the Known Variables into the Expression

Substitute \(D = 1000\) and \(A = 8\) into the expression to get \(\frac{D A}{A+12} = \frac{1000 * 8}{8 + 12}\)

Step 3: Simplify the obtained Expression

The expression simplifies to \(\frac{8000}{20}\)

Step 4: Perform the Division

Finally, carry out the division to get the child's dosage. The result of \(\frac{8000}{20}\) equals 400.

Key Concepts

Dosage Calculation

Dosage calculation is crucial in ensuring medications are both safe and effective, especially for children. Child dosages often require special consideration because children are not simply smaller adults; they have different physiological responses and metabolic rates. Using rational expressions, such as \(\frac{D A}{A+12}\), provides a structured way to determine appropriate medication dosages for children based on their age and the standard adult dosage.

• \(D\) is the adult dosage.
• \(A\) is the child's age.
• The formula adjusts the dosage based on these factors.

For instance, if an adult dosage \((D)\) is 1000 mg, substituting a child's age \((A)\) of 8 years into the formula leads to calculations ensuring the child receives a suitable amount. This rational expression helps ensure that healthcare professionals provide pediatric patients with dosages that are tailor-fit to their needs, minimizing risks and maximizing therapeutic efficacy.

Substitution in Algebra

Substitution is a fundamental concept in algebra that makes problem-solving more straightforward. It's like inserting known values into equations to simplify and solve them quickly. In our example of dosage calculation, we identified the known variables: \(D = 1000\) and \(A = 8\). Substituting these into the expression \(\frac{D A}{A+12}\) allowed us to proceed with an accurate calculation.Substitution helps transform complex-looking rational expressions into simpler numerical equations that are much easier to handle. This method makes it possible to explore and solve problems involving any variable or set of variables across different contexts, not just in medicinal dosage scenarios but also in geometry, physics, and other areas of mathematics. By replacing variables, you change an abstract expression into a concrete problem that can be tackled with basic arithmetic.

Problem-Solving Steps

Following structured problem-solving steps ensures efficiency and accuracy when dealing with mathematical problems like dosage calculations. Here is an outline of how to approach such problems:1. **Identify Known Values:** Clarifying what values and variables are given in a problem helps streamline the solving process. For example, knowing \(D\) and \(A\) right away simplifies the task. 2. **Substitute Values Into the Equation:** Inserting these known values into the rational expression or equation helps in simplifying it and making the solution clear.3. **Simplify the Expression:** Once values are substituted, simplify the expression to its most basic form. This may involve performing multiplication, division, or other arithmetic operations.4. **Carry Out the Arithmetic:** Finally, complete any needed arithmetic, such as division or multiplication, to find the solution. In our example, dividing \(\frac{8000}{20}\) confirmed the child's dosage calculation as 400 mg.Breaking down a problem into these manageable steps not only boosts problem-solving ability but also reinforces the understanding of the concepts within rational expressions.