Problem 63 - Algebra and Trigonometry

Question

If the recommended adult dosage for a drug is \(D(\text { in mg), then to determine the appropriate dosage }\) \(c\) for a child of age \(a,\) pharmacists use the equation $$c=0.0417 D(a+1)$$ Suppose the dosage for an adult is 200 \(\mathrm{mg}\) . (a) Find the slope. What does it represent? (b) What is the dosage for a newborn?

Step-by-Step Solution

Answer

The slope is 8.34, meaning the dosage increases by 8.34 mg per year. A newborn's dosage is 8.34 mg.

Step 1: Understand the Equation

The equation used to determine the child's dosage is \(c = 0.0417 D(a+1)\). This equation implies that a child's dosage \(c\) depends on the adult dosage \(D\) and the child's age \(a\). The constant \(0.0417\) is the factor used to adjust the adult dosage for a child, taking the age into account.

Step 2: Identify the Slope in the Equation

By comparing the child dosage equation \(c=0.0417 D(a+1)\) to the form of a linear equation \(y = mx + b\) where \(m\) is the slope, we identify the slope as \(0.0417D\). This slope represents the rate at which the child's dosage increases for each additional year of age after adjustment based on the adult's dose.

Step 3: Calculate the Slope for Given Adult Dosage

Substitute \(D = 200\) into the slope: \(0.0417 \times 200 = 8.34\). The slope 8.34 means that for every additional year in the child's age, the dosage increases by 8.34 mg, assuming \(D\) stays constant.

Step 4: Calculate Dosage for a Newborn

A newborn can be considered to be at age \(a = 0\). Substitute \(a = 0\) and \(D = 200\) into the equation: \(c = 0.0417 \times 200 \times (0+1)\). So, \(c = 0.0417 \times 200 = 8.34\).

Step 5: Confirm Units and Interpretation

Finally, ensure the dosage for the newborn is expressed in milligrams, confirming correctness with \(c = 8.34 \, \text{mg}\). This is the newborn's dose when an adult's recommended dosage is 200 mg.

Key Concepts

Pharmacology

Pharmacology is the branch of medicine that focuses on the interaction between drugs and biological systems. One key aspect of pharmacology involves determining the correct dosages of medications for different individuals, ensuring both efficacy and safety. Dosage calculations take into account various factors such as age, weight, and health conditions. For children, dose adjustments are particularly important because their bodies process drugs differently than adults.
In the context of this exercise, pharmacological principles are applied to ensure proper dosing for children using the equation \(c = 0.0417 D(a+1)\). This formula allows pharmacists to customize dosages based on age, offering a systematic approach to medication administration. Understanding how to adjust adult dosages to suitable pediatric levels is crucial in pharmacology to avoid adverse effects and achieve therapeutic goals.

Mathematical Modeling

Mathematical modeling involves using mathematical equations to represent real-world systems. In pharmacology, these models help predict how different variables affect drug dosage and response. This exercise demonstrates the application of a simple mathematical model to determine child dosage based on adult recommendations.
By analyzing the relationship between the child's age \(a\) and the recommended dosage \(D\), the model allows for accurate dose calculation using the provided linear equation. Models like this simplify complex biological processes and assist in making data-driven decisions. Mathematical modeling is valuable because it provides a structured way to visualize and analyze how changes in various factors impact outcomes, in this case, drug dosage for children.

Linear Equations

Linear equations describe relationships where there is a constant rate of change between two variables. In the exercise, the equation \(c = 0.0417 D(a+1)\) is linear. Here, the child's dosage \(c\) is the dependent variable, while \(a\), the child's age, is the independent variable. The expression \(0.0417D\) represents the slope, indicating how the dose changes as the child's age increases.
In practical terms, the slope tells us how much the dosage will increase for each additional year of the child's age. With the given adult dosage of 200 mg, the slope is computed as 8.34 mg per year. Linear equations are widely used in various fields including pharmacokinetics, where they help describe proportional relationships and simplify complex biological data into manageable forms.

• They provide clarity by showing direct dependencies between variables.
• They allow us to predict outcomes based on known data points.

Linear equations thus become powerful tools in both mathematics and applied sciences.