Problem 126
Question
Dosages. The amount of medicine a patient should take is often proportional to his or her weight. If a patient weighing 83 kilograms needs 150 milligrams of medicine, how much will be needed by a person weighing 99.6 kilograms?
Step-by-Step Solution
Verified Answer
A person weighing 99.6 kg needs approximately 180 milligrams of medicine.
1Step 1: Determine the Proportionality Constant
The first step is to determine the proportionality constant for the dosage based on the given weight and dosage. We know that a patient weighing 83 kg needs 150 mg of medicine. The proportionality constant, \( k \), can be found using the formula: \( k = \frac{150}{83} \). Calculate this value to find \( k \).
2Step 2: Set Up the Equation for the New Patient
Next, using the proportionality constant found, set up the equation to find the required dosage for a patient weighing 99.6 kg. The equation is \( ext{Dosage} = k imes ext{Weight} \). Substitute the known values into the equation.
3Step 3: Calculate the New Dosage
Substitute \( k \) and the new weight of 99.6 kg into the equation to calculate the needed dosage. The calculation is: \( ext{Dosage} = \frac{150}{83} imes 99.6 \). Compute this to find the required dosage for the patient.
4Step 4: Conclude with the Final Dosage Amount
Upon calculating, you will have the specific dosage amount required for the patient weighing 99.6 kilograms. This concludes the solution.
Key Concepts
Dosage CalculationProportion ConstantWeight and Dosage Relationship
Dosage Calculation
Calculating the correct medicine dosage based on a person's weight is a crucial skill in both healthcare and pharmaceutical settings. To start, identify the dosage provided to a known weight. For instance, if a person at 83 kilograms is prescribed 150 milligrams, this forms the base for our calculation.
The main aim is to adjust the dosage to suit different weights while maintaining the same effectiveness. You achieve this by employing mathematical concepts like proportionality. Essentially, you'll be scaling the original dosage up or down according to the new weight. This ensures that all patients receive the appropriate amount of medicine necessary for their size.
Following the steps systematically, you initially determine the proportional constant using existing dosage and weight information. Subsequently, you create an equation making use of this constant to define the dosage for the new weight.
The main aim is to adjust the dosage to suit different weights while maintaining the same effectiveness. You achieve this by employing mathematical concepts like proportionality. Essentially, you'll be scaling the original dosage up or down according to the new weight. This ensures that all patients receive the appropriate amount of medicine necessary for their size.
Following the steps systematically, you initially determine the proportional constant using existing dosage and weight information. Subsequently, you create an equation making use of this constant to define the dosage for the new weight.
Proportion Constant
The proportion constant, often denoted as \( k \), plays a central role in dosage calculations, ensuring that the medicine dosage scales appropriately with weight changes. This constant is derived from the known relationship between a specific weight and its corresponding dosage.
To find \( k \), use the formula:
Once determined, \( k \) is then used to compute dosages for other weights, ensuring consistency in dosage potentially across varied patient profiles.
To find \( k \), use the formula:
- \( k = \frac{\text{Dosage}}{\text{Weight}} \)
Once determined, \( k \) is then used to compute dosages for other weights, ensuring consistency in dosage potentially across varied patient profiles.
Weight and Dosage Relationship
The relationship between weight and the appropriate medication dosage is a fundamental principle in pharmacology. It's imperative to grasp that the dosage should proportionally increase with the patient's weight. This supports the idea that heavier individuals require larger doses to achieve the same therapeutic effect as smaller individuals.
If you understand that the weight-dosage connection is a direct proportion, it becomes clearer why this relationship is linear. For every unit increase in weight, the dosage increases by a consistent amount, determined by the proportion constant \( k \).
In practical application, pharmacists and doctors rely on these calculations to ensure efficacy and safety. By using weight as a key variable, medical professionals can tailor treatments to individual needs while maintaining a standardized approach across diverse patient categories.
If you understand that the weight-dosage connection is a direct proportion, it becomes clearer why this relationship is linear. For every unit increase in weight, the dosage increases by a consistent amount, determined by the proportion constant \( k \).
In practical application, pharmacists and doctors rely on these calculations to ensure efficacy and safety. By using weight as a key variable, medical professionals can tailor treatments to individual needs while maintaining a standardized approach across diverse patient categories.
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