Question

(a) How many liters of wine can be held in a wine barrel whose capacity is 31 gal? (b) The recommended adult dose of Elixophyllin", a drug used to treat asthma, is \(6 \mathrm{mg} / \mathrm{kg}\) of

Short answer

The wine barrel can hold approximately 117 liters of wine, and a 50-kg person should be prescribed 300 mg of Elixophyllin.

Step-by-step solution

Conversion of gallons to liters

First, we need to convert the capacity of the wine barrel from gallons to liters. To do this, we need to know the conversion factor between gallons and liters: 1 gallon is equal to 3.78541 liters.

Calculate the capacity in liters

Now that we have the conversion factor, we can calculate the capacity of the wine barrel in liters: \[Capacity_{Liters} = Capacity_{Gallons} \times ConversionFactor\] \[Capacity_{Liters} = 31 \times 3.78541\]

Calculate the total capacity in liters

Multiply the number of gallons by the conversion factor to obtain the total capacity in liters: \[Capacity_{Liters} = 117.24771\]

Round the result

Round the capacity to the nearest whole number: \[Capacity_{Liters} \approx 117\] So, the wine barrel can hold approximately 117 liters of wine.

Calculate the Elixophyllin dose

For part (b) of the problem, we know that the recommended adult dose of Elixophyllin is \(6\, \frac{mg}{kg}\) of body weight. We need to calculate the prescribed dosage for a person weighing 50 kg: \[Dosage_{mg} = Dose_{\frac{mg}{kg}} \times Weight_{kg}\]

Find the prescribed dosage

Multiply the given dose by the person's weight in kg: \[Dosage_{mg} = 6 \times 50\]

Calculate the total dosage

Perform the calculation to find the total dosage in milligrams: \[Dosage_{mg} = 300\] So, a 50-kg person should be prescribed 300 mg of Elixophyllin.

Key concepts

Liters to Gallons Conversion

Converting between liters and gallons is a common task in chemistry, especially when dealing with different measurement systems. When we need to convert gallons to liters, it's vital to use the correct conversion factor. For instance, 1 gallon is exactly 3.78541 liters. This conversion is crucial for ensuring that measurements are accurate in scientific calculations and everyday scenarios.

Let's break it down with an example: if you have a wine barrel with a capacity of 31 gallons, and you want to find out how many liters it holds, you simply multiply the number of gallons by the conversion factor:

• \[Capacity_{Liters} = 31 \times 3.78541 = 117.24771 \]
• After performing the calculation, you might need to round to the nearest whole number for simplicity, leading to approximately 117 liters.

This process underscores the importance of unit conversion in scientific work, allowing us to work seamlessly across different measurement systems.

Dosage Calculation

Dosage calculation is a critical concept, especially in medical fields where precise amounts of medication are crucial for effective treatment. Understanding how to calculate the correct dosage based on a patient's weight ensures patient safety and treatment efficacy.

Consider the example where an asthma medication called Elixophyllin is prescribed at a dosage of 6 milligrams per kilogram of body weight. If you need to determine the appropriate dose for a 50 kg adult, you'd multiply the dosage per kilogram by the patient's body weight.

• \[Dosage_{mg} = 6 \times 50 = 300 \]

The total dosage for a 50 kg person would be 300 mg.

This straightforward calculation demonstrates the necessity of understanding units and conversions in healthcare practices, ensuring that patients receive the correct medication dosage.

Metric System Conversion

The metric system is prevalent in scientific research because of its simplicity and universality. It is based on powers of ten, which makes conversions within the metric system straightforward and easy to understand.

A helpful way to remember the key units in the metric system is through prefixes like kilo, centi, and milli, which indicate the scale of measurement. For example, 1 kilometer is 1000 meters, while 1 millimeter is 0.001 meters.

The ease of multiplying or dividing by factors of 10 when converting makes the metric system a favored choice in science. Moreover, these units are standardized globally, which reduces confusion and errors in measurements.

• This standardization streamlines scientific communication and data sharing across countries and disciplines.

Mastery of metric conversions is crucial, as seen in chemistry, where precise measurements and conversions can significantly impact experimental results.