Problem 66 - College Algebra

Question

Doctors use the rational expression $$\frac{D A}{A+12}$$ to determine the dosage of a drug prescribed for children. In this expression, \(A=\) child's age, and \(D=\) adult dosage. What is the difference in the child's dosage for a 7-year-old child and a 3 -year-old child? Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the rational expression.

Step-by-Step Solution

Answer

\(\frac{16D}{95}\)

Step 1: Determine the dosage for a 7-year-old child

Substitute \(A = 7\) into the rational expression \(\frac{DA}{A+12}\) to get the dosage for a 7-year-old child, which becomes \(\frac{7D}{7+12} = \frac{7D}{19}\)

Step 2: Determine the dosage for a 3-year-old child

Substitute \(A = 3\) into the rational expression \(\frac{DA}{A+12}\) to get the dosage for a 3-year-old child, which becomes \(\frac{3D}{3+12} = \frac{3D}{15}\)

Step 3: Calculate the difference

Subtract the dosage for a 3-year-old child from the dosage for a 7-year-old child to compute the difference, which will be \(\frac{7D}{19} - \frac{3D}{15}\)

Step 4: Find an equivalent rational expression

To subtract these rational expressions, first find a common denominator (in this case, 285, which is the product of 19 and 15), then subtract the numerators to get \(\frac{(7D \cdot 15) - (3D \cdot 19)}{285} = \frac{105D-57D}{285} = \frac{48D}{285}\)

Step 5: Simplify the rational expression

We can simplify this expression by dividing both the numerator and the denominator by 3 to get \(\frac{16D}{95}\)

Step 6: Interpret the answer in terms of the variables in the original rational expression

The result, \(\frac{16D}{95}\), represents the difference in dosage for an adult drug prescribed to a 7-year-old child and a 3-year-old child. It means that a 7-year-old child will be prescribed \(\frac{16D}{95}\) more of the adult dosage (D) than a 3-year-old child.

Key Concepts

Dosage Calculation

Understanding how to calculate dosages is crucial when prescribing medicines, especially for children. The dosage of a medication is typically based on various factors, including the age of the child and the adult dosage amount. Doctors use rational expressions to fine-tune these calculations. A typical rational expression for children's dosage is given by \( \frac{DA}{A+12} \). Here, \( A \) stands for the child's age, while \( D \) indicates the adult dosage.

To calculate dosages:

• Insert the child's age into the variable \( A \).
• Perform the division in the expression to get the result.

This method ensures a proportional amount of the medication is administered, taking into account both the standard adult dosage and the child's age-related needs. Ensuring the right dosage is calculated correctly is vital for the safety and efficacy of the medication.

Children's Medication Dosage

When it comes to children's medication dosage, it is important to adjust the dosage according to specific parameters rather than using a 'one-size-fits-all' approach. The rational expression \( \frac{DA}{A+12} \) is a formula used by doctors to tailor the adult dosage to suit children's needs.

To better understand this concept:

• Identify the adult dosage \( D \).
• Use the child's specific age to adjust the dosage formula.
• Perform the necessary calculations to determine the exact dosage.

In the exercise, we learned that for a 7-year-old, the calculation will yield \( \frac{7D}{19} \), while for a 3-year-old, it results in \( \frac{3D}{15} \). This difference emphasizes how the dosage must change as the age varies, ensuring the child receives an age-appropriate dosage.

Age and Drug Dosage

Age plays a critical role in determining the appropriate drug dosage for children. As children grow, their bodies can handle larger amounts of medication. The rational expression \( \frac{DA}{A+12} \) effectively shows how age affects drug dosage.

In the example given:

• A 7-year-old child receives \( \frac{7D}{19} \), reflecting their ability to handle a larger dose.
• A 3-year-old child receives \( \frac{3D}{15} \), with less ability to metabolize larger doses safely.

The difference between these dosages, calculated as \( \frac{16D}{95} \), illustrates the incremental increase as children age. Using such rational expressions ensures that dosages safely increase to meet children's developmental needs while preventing overdose risks. This method underscores the importance of adjusting dosages according to individual physiological capabilities.