Problem 59 - 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 300 milligrams. What is that child's age? Round to the nearest year.

Step-by-Step Solution

Answer

The child's age is approximately 5 years.

Step 1: Identifying Knowns and Unknowns

Here, the quantities of interest are given as: Adult dosage (D) = 1000 milligrams, Child's dosage (C) = 300 milligrams, and the child's age (A) is what we are trying to find. The original formula is \(C=\frac{D A}{A+12}\). To solve for A, the formula should be rearranged.

Step 2: Rearranging The Formula for A

By cross multiplying and simplifying, the formula can be rearranged as \(A=\frac{12C}{D-C}\).

Step 3: Substitute the Given Values

Plugging the given values of adult dosage (D) and child's dosage (C) into the rearranged formula gives \(A=\frac{12 * 300}{1000 - 300}\).

Step 4: Calculate The Child's Age

Finally, by performing the calculations, the result is \(A \approx 5 \) years when rounded to the nearest year.

Key Concepts

Young's Rule

Young's Rule is a handy formula used in pharmacology to approximate appropriate drug dosages for children based on their age. This is particularly important because the dosage required naturally changes with age and weight, and using adult dosages can be unsafe for children.

The formula is given as \( C=\frac{D A}{A+12} \), where:

• \( C \) represents the child's dosage.
• \( D \) represents the adult dosage.
• \( A \) represents the child's age.

Using this rule helps ensure that the medication is safe for use by adjusting according to a child's maturity level compared to adults. Understanding how to rearrange and use the formula correctly is essential in safely administering medications. In problems like ours, the goal is often to isolate and find a specific unknown, such as a child’s age, by manipulating the formula.

Dosage Calculation

Calculating the right drug dosage is vital for safety and efficacy. Young's Rule makes this process a little simpler by providing a method to find out either the necessary dosage or a child's age for a given dosage.

In the exercise, we started by identifying the known values, which were:

• \( D \), the adult dosage, is 1000 milligrams.
• \( C \), the child's dosage, is 300 milligrams.
• \( A \), the child's age, is the unknown we need to calculate.

Rearranging the formula for \( A \), we have \( A=\frac{12C}{D-C} \).

Then, by substituting the actual known values into this new formula, we calculate the child's age.Substituting into the formula, \( A=\frac{12 \times 300}{1000 - 300} \), performs mathematical operations to find \( A \). When solved, we find \( A \) is approximately 5 years. Calculations like these should always be checked carefully to avoid errors.

Problem Solving

Successfully applying a formula like Young's Rule involves clear problem-solving steps. It's essential to follow a structured approach to deal with any algebraic equation effectively.

Here's how you can approach such problems:

• **Identify knowns and unknowns:** Start by writing down what you know and what you need to find from the problem.
• **Rearrange formulas as needed:** Rearrange the equation to isolate the unknown you need to solve for. For Young’s Rule, this involved algebraic manipulation to solve for the child’s age \( A \).
• **Substitute known values:** Once the equation is rearranged, plug in the given values you know from your problem setup.
• **Perform the calculations carefully:** Work through the math, checking each step to ensure accuracy.
• **Verify your solution:** After calculating, it helps to reconsider your results or even plug them back into the original equation to ensure they fit correctly.

Following these problem-solving steps ensures accuracy and builds confidence in tackling similar dosage calculation problems. It helps break down what seems complex into manageable tasks.