Problem 61 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Thomas Young has suggested the following rule for calculating the dosage of medicine for children 1 to 12 yr old. If \(a\) denotes the adult dosage (in milligrams) and if \(t\) is the child's age (in years), then the child's dosage is given by $$ D(t)=\frac{a t}{t+12} $$ Suppose the adult dosage of a substance is \(500 \mathrm{mg}\). Find an expression that gives the rate of change of a child's dosage with respect to the child's age. What is the rate of change of a child's dosage with respect to his or her age for a 6-yrold child? A 10 -yr-old child?

Step-by-Step Solution

Answer

The rate of change of a child's dosage with respect to age is given by the expression \(\frac{6000}{(t+12)^{2}} mg/year\). For a 6-year-old child, this rate is approximately \(18.52 mg/year\), and for a 10-year-old child, it is approximately \(12.40 mg/year\).

Step 1: Rewrite the dosage formula for an adult dosage of 500 mg

The given formula for a child's dosage based on their age and adult dosage is: \(D(t)=\frac{a t}{t+12}\) Given that the adult dosage of a substance is 500 mg, we have: \[D(t)=\frac{500t}{t+12}\]

Step 2: Calculate the derivative of the dosage function with respect to age

To find the rate of change, we need to compute the derivative of \(D(t)\) with respect to \(t\). We can use the quotient rule. \[\frac{dD}{dt} = \frac{(t+12)(500) - (500t)(1)}{(t+12)^{2}}\]

Step 3: Simplify the derivative of the dosage function

Now, simplify the result from the previous step: \[\frac{dD}{dt} = \frac{500t+6000 - 500t}{(t+12)^{2}} = \frac{6000}{(t+12)^{2}}\]

Step 4: Evaluate the derivative as t = 6 for a 6-year-old child

Now, we plug in the value of \(t = 6\) to find the rate of change of the dosage for a 6-year-old child: \[\frac{dD}{dt}(6) = \frac{6000}{(6+12)^{2}} = \frac{6000}{18^{2}} = \frac{6000}{324} = 18.52 mg/year\]

Step 5: Evaluate the derivative as t = 10 for a 10-year-old child

Similarly, we plug in the value of \(t = 10\) to find the rate of change of the dosage for a 10-year-old child: \[\frac{dD}{dt}(10) = \frac{6000}{(10+12)^{2}} = \frac{6000}{22^{2}} = \frac{6000}{484} = 12.40 mg/year\] To summarize, the rate of change of a child's dosage with respect to his or her age is given by the expression \(\frac{6000}{(t+12)^{2}} mg/year\). For a 6-year-old child, this rate is approximately \(18.52 mg/year\), and for a 10-year-old child, it is approximately \(12.40 mg/year\).

Key Concepts

Dosage Calculation Formula

When it comes to determining the amount of medication appropriate for children, healthcare professionals often rely on specific formulas. One such formula is Young's rule, which helps in calculating the dosage of medicine for children aged between 1 to 12 years. The formula is:
\[\begin{equation}D(t)=\frac{a t}{t+12}\end{equation}\]
In this formula, \(D(t)\) is the child's dosage, \(a\) represents the adult dosage, and \(t\) is the age of the child in years. The beauty of this formula lies in its simplicity and direct consideration of the child's age as a factor in determining the dosage. This ensures that younger children, with their smaller bodies and developing systems, receive less medication than an adult would.
For instance, if the adult dosage is 500 mg, you can easily calculate the correct dosage for any child's age by substituting the age into the formula. It's a practical application of mathematics in pharmacology that streamlines the dosage adjustment process for pediatric patients.

Quotient Rule Differentiation

Understanding Quotient Rule

Differentiation in calculus is a tool used to calculate the rate of change. For functions formulated as a quotient, like our dosage formula, the quotient rule becomes particularly useful. It provides a method to differentiate functions that are ratios of two other functions.
Here is the quotient rule in a generic form:\[\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\]
Applied to our dosage calculation, where we consider the child's age relative to the dosage, the quotient rule allows us to compute how the dosage should change as the child ages. This derivative, then, is the rate of change of the medicine dosage—essentially how much more or less medicine the child would need with each passing year.

Age-Based Dosage Adjustment

Factors Influencing Age-Based Dosage

When adjusting medication dosages based on a patient's age, various physiological factors are taken into account, such as metabolic rate, liver and kidney function, and body size. Young's rule is a formulaic reflection of these considerations. It encapsulates the complex relationship between age and medication dosage into a simpler mathematical expression. This makes it possible to estimate the appropriate dosage for children without the need for trial and error or risky guesswork.
By using the dosage formula and its derived rate of change, healthcare providers can predict how a child's medication needs will evolve with age. Age-based adjustments are crucial because children are not simply 'small adults'—their bodies process substances differently, and as they grow, the variations in these processes must be factored into the medication they receive.

Applied Mathematics in Pharmacology

Pharmacology is rife with complex calculations, and rightfully so; precise dosages can mean the difference between effective treatment and potential harm. Applied mathematics provides the tools necessary to navigate these complexities.
The use of algebra, calculus, and other mathematical disciplines allows for the creation of models like Young's rule, which we've explored. Moreover, mathematical concepts such as the rate of change give insight into how dosages should adapt over time, ensuring safe and effective pharmacological practices. Medication dosing formulas, age-based adjustments, and differential calculus all exemplify applied mathematics in this field, and understanding them is pivotal for healthcare professionals to administer drugs safely and effectively.