Problem 72 - Intermediate Algebra

Question

Use a proportion to solve each problem. Recommended Dosages. The recommended child's dose of the sedative hydroxine is 0.006 gram per kilogram of body mass. Find the dosage for a 30 -kg child in grams and in milligrams.

Step-by-Step Solution

Answer

The dosage for a 30-kg child is 0.18 grams or 180 milligrams.

Step 1: Understand the Proportion

The given information states that the recommended dosage of hydroxine is 0.006 grams per kilogram of body weight. Therefore, the proportion can be set up as follows: \[ \text{Dosage for 1 kg} = 0.006 \text{ grams} \] This will be used to calculate the dosage for a 30-kg child.

Step 2: Set Up the Equation

Use the proportion to determine the necessary dosage for the entire body mass of the child. Let \( x \) represent the dosage for a 30-kg child in grams. The equation is: \[ 30 \text{ kg} \times 0.006 \text{ grams/kg} = x \]

Step 3: Solve for Dosage in Grams

Multiply the mass of the child by the dosage per kilogram to find \( x \). The calculation is: \[ x = 30 \times 0.006 = 0.18 \text{ grams} \] This means that the child's dosage should be 0.18 grams.

Step 4: Convert Dosage to Milligrams

To convert grams to milligrams, use the conversion factor that 1 gram equals 1000 milligrams. Therefore, the dosage in milligrams is: \[ 0.18 \text{ grams} \times 1000 = 180 \text{ milligrams} \] This means that the dosage is 180 milligrams.

Key Concepts

Dosage Calculation

Calculating the right dosage for medication is crucial, especially when it comes to children. Dosage calculation involves determining how much of a substance should be administered based on specific criteria. In the case of the sedative hydroxine, the criteria provided is that the dosage is 0.006 grams per kilogram of body mass. To find the correct dosage for a child weighing 30 kilograms, a simple multiplication is conducted:

• Multiply 30 kilograms by 0.006 grams per kilogram.

This results in the calculation:

• \( x = 30 \times 0.006 = 0.18 \text{ grams} \)

Understanding dosage calculation ensures safety and effectiveness in medication administration.

Unit Conversion

Once the dosage is calculated in grams, the next step often involves converting it into another unit. Unit conversion in pharmacology is common, as prescriptions might need to meet different standards or practices such as milligrams, which are a smaller unit than grams. To convert from grams to milligrams:

• Use the conversion factor where 1 gram is equal to 1000 milligrams.
• Multiply the dosage in grams by this factor.

For the dosage of 0.18 grams, the conversion to milligrams becomes:

• \( 0.18 \text{ grams} \times 1000 = 180 \text{ milligrams} \)

Practicing unit conversion helps in accurate dose administration, thus preventing underdose or overdose of medication.

Mathematical Equations

Mathematical equations are at the heart of dosage calculations and unit conversions. These equations are often simple multiplications or proportional setups that guide the calculation process.

• The proportion provided in the exercise is \( 0.006 \text{ grams} \) per \( 1 \text{ kilogram} \).

To solve for the total dosage, set up an equation:

• \( 30 \text{ kg} \times 0.006 \text{ grams/kg} = x \)

The solution is the product of these variables, resulting in the correct dosage in grams. Mathematical equations like this one help us apply logical thinking and problem-solving skills to real-life scenarios, ensuring that medication calculations are handled precisely.