Problem 49 - Calculus for Biology and Medicine

Question

To convert the volume of a liquid measured in ounces to a volume measured in liters, we use the fact that 1 liter equals \(33.81\) ounces. Denote by \(x\) the volume measured in ounces and by \(y\) the volume measured in liters. Assume a linear relationship between these two units of measurements. (a) Find the equation relating \(x\) and \(y\). (b) A typical soda can contains 12 ounces of liquid. How many liters is this?

Step-by-Step Solution

Answer

Equation: \(y = \frac{x}{33.81}\); 12 ounces is approximately 0.3548 liters.

Step 1: Identify the Conversion Factor

The conversion factor between ounces and liters is given as \(1 \text{ liter} = 33.81 \text{ ounces}\). This means \(x\) ounces is converted to liters using the equation \(y = \frac{x}{33.81}\).

Step 2: Establish the Equation

Using the conversion factor, the linear equation relating \(x\) and \(y\) can be written as \(y = \frac{1}{33.81}x\). This equation expresses the volume in liters \(y\) in terms of the volume in ounces \(x\).

Step 3: Solve for a 12-Ounce Soda Can

Given \(x = 12\) ounces for a typical soda can, substitute \(x = 12\) into the equation \(y = \frac{1}{33.81}x\).

Step 4: Calculation

Calculate \(y\) by evaluating \(y = \frac{1}{33.81} imes 12\). This gives \(y \approx 0.3548\) liters.

Key Concepts

Understanding Linear Equations

Linear equations are foundational in mathematics and are particularly useful for converting units, such as finding the relationship between ounces and liters. An equation is considered linear if it graphically represents a straight line. This usually occurs when the relationship between two variables is proportional. In our exercise, the relationship between the volume in ounces and the volume in liters is linear.

We define this relationship by a linear equation:

• In this case, the equation is given as \(y = \frac{1}{33.81}x\)
• This implies, for each value of \(x\) (ounces), we can find \(y\) (liters) by multiplying \(x\) by \(\frac{1}{33.81}\).

Linear equations simplify the process of conversion as they provide a straightforward formula to convert one unit to another without complex calculations.

The Role of Conversion Factors

A conversion factor is crucial when dealing with different units of measurement. It acts as a bridge between these units, allowing us to convert a value from one unit to another. In our exercise, the conversion factor between ounces and liters is given as \(1\) liter equals \(33.81\) ounces.

Here's how it works:

• The conversion factor of \(\frac{1}{33.81}\) is used to change ounces to liters.
• We multiply the volume in ounces by this factor to find the equivalent volume in liters.

Understanding and using conversion factors properly ensures accuracy during the conversion process. It's important to remember that conversion factors usually come from established scientific measurements and should be used as intended to maintain consistency and correctness.

Basics of Volume Measurement

Volume measurement refers to the quantity of three-dimensional space occupied by a liquid, solid, or gas. In everyday use, volume can be measured in various units like liters, gallons, or ounces depending on the context and region.

In the context of converting ounces to liters, understanding volume measurement is key:

• An ounce is a unit of volume commonly used in the United States for liquids.
• A liter is a metric unit of volume that is widely used around the world and in scientific contexts.

For example, when measuring a typical soda can containing \(12\) ounces, by converting it to liters, we are aligning with a universal standard which can be useful for recipes, scientific purposes, or international communication. By understanding different volume measurements, we can easily operate in both domestic and international settings.