Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Cowling's rule is a method for calculating pediatric drug dosages. 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 $$c=\left(\frac{t+1}{24}\right) a$$ a. Solve the equation for \(t\) in terms of \(a\) and \(c\). b. If the adult dose of a drug is \(500 \mathrm{mg}\) and a child received a dose of \(125 \mathrm{mg}\), how old was the child?

Step-by-Step Solution

Answer

The child's age \(t\) in terms of the adult dosage \(a\) and child's dosage \(c\) is given by the equation \(t = \frac{24c}{a} - 1\). Substituting the given values, \(a = 500 \mathrm{mg}\) and \(c = 125 \mathrm{mg}\), we find that the child is 5 years old.

Step 1: Rewrite the equation

Start by writing the given formula to isolate \(t\): \(c = \frac{t + 1}{24} a\)

Step 2: Multiply both sides by 24

To get rid of the fraction, multiply both sides by 24: \(24c = (t + 1)a\)

Step 3: Divide both sides by a

Now, divide both sides of the equation by \(a\) to isolate \(t + 1\): \(\frac{24c}{a} = t + 1\)

Step 4: Subtract 1 from both sides

Finally, subtract 1 from both sides to isolate \(t\): \(t = \frac{24c}{a} - 1\) #b. If the adult dose of a drug is \(500 \mathrm{mg}\) and a child received a dose of \(125 \mathrm{mg}\), how old was the child?#

Step 1: Substitute the given values

Given \(a = 500 \mathrm{mg}\) and \(c = 125 \mathrm{mg}\), we can substitute these values into the equation we found in part a: \(t = \frac{24 \cdot 125}{500} - 1\)

Step 2: Simplify the expression

Simplify the expression to find the value of \(t\): \(t = \frac{3000}{500} - 1\)

Step 3: Calculate the age

Divide the numerator by the denominator and subtract 1 to find the age of the child: \(t = 6 - 1\)

Step 4: Write the final answer

Calculate the final answer: \(t = 5\) Therefore, the child is 5 years old.

Key Concepts

Pediatric Drug Dosages

Understanding how to calculate the correct medication dose for children is critical, as their bodies process drugs differently than adults. The standard dosage for an adult may be dangerous for a child, and so pediatric dosages must be adjusted according to factors such as the child's age and weight.

Cowling's rule is one specific formula that helps determine the appropriate dosage for a child based on their age. The formula establishes a direct proportion between the adult's dosage and the child's dosage by incorporating a ratio that accounts for the age of the child.

When using Cowling's rule, it's crucial to consider that it is just one of several available methods, and healthcare providers may use different rules for different medications or under varying circumstances. Additionally, this rule doesn't consider a child's weight, which can be a significant factor in medication dosing. Thus, Cowling's rule should be used with caution and under professional guidance.

Solving for Unknown Variables

A common problem in algebra is solving for an unknown variable. This applies to various real-life applications, including medicine dosage calculations like we see with Cowling's rule. The process involves algebraic manipulation to isolate the variable on one side of the equation.

In Cowling's rule, the unknown variable represents the child's age when we're given the adult and child dosages. To find the child’s age, we rearrange the equation step by step. Multiplying by common denominators, dividing by known quantities, and subtracting constants are typical steps in algebra used to isolate the variable.

Successfully solving for unknown variables often requires methodical thinking and practice with algebraic principles. It's an indispensable skill in many scientific and mathematical contexts beyond pharmacology.

Medicine Dosage Calculation

Calculating the correct medicine dosage is not only crucial for patient safety but also for the efficacy of treatment. Cowling's rule is an example of these calculations, which are often used in pediatric medicine.

Following the steps to solve this type of problem, we start with the standard equation and incorporate the known values (adult dosage and child's received dose). Through arithmetic operations such as multiplication and division, we adjust the adult dosage to find the safe dosage for a child of a specific age.

The process demonstrates the importance of accuracy in medicine dosage calculation, as even small errors can result in significant consequences for patient health. Pharmacists and healthcare providers use these calculations regularly, reaffirming the need for precision and careful double-checking of work.