Problem 2 - University Physics with Modern Physics

Question

According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions \(1 \mathrm{L}=1000 \mathrm{cm}^{3}\) and \(1 \mathrm{in.}=2.54 \mathrm{cm},\) express this volume in cubic inches.

Step-by-Step Solution

Answer

The volume of the salad dressing bottle is approximately 28.86 cubic inches.

Step 1: Convert Liters to Cubic Centimeters

Start with the given volume of 0.473 liters. We know that 1 liter equals 1000 cubic centimeters. So, we convert liters to cubic centimeters using this conversion: \(0.473 \, \text{L} \times 1000 \, \text{cm}^3/\text{L} = 473 \, \text{cm}^3\).

Step 2: Convert Cubic Centimeters to Cubic Inches

Next, we convert cubic centimeters to cubic inches. We are given that \(1 \, \text{in.} = 2.54 \, \text{cm}\). Therefore, \(1 \, \text{cm} = \frac{1}{2.54} \, \text{in.}\). To convert cubic measurements, cube the conversion factor: \(1 \, \text{cm}^3 = (\frac{1}{2.54})^3 \, \text{in}^3 = \frac{1}{16.387064} \, \text{in}^3\).

Step 3: Perform the Final Conversion

Use the equation from Step 2 to convert 473 cubic centimeters to cubic inches: \(473 \, \text{cm}^3 \times \frac{1}{16.387064} \, \text{in}^3/\text{cm}^3 \approx 28.86 \, \text{in}^3\).

Key Concepts

Cubic Measurement Conversions

Cubic measurement conversions are used when you want to convert volumes from one unit to another, particularly in three-dimensional space. These conversions are essential because they allow you to express the volume in different units, such as shifting from centimeters to inches or liters to milliliters, depending on the requirement.
When converting units for volume:

• Always ensure that you cube the linear conversion factor. This means if you know the conversion from centimeters to inches, like 1 cm = 0.393701 inches, for cubic conversion, cube the factor: (1 cm³ = 0.393701³ in³).
• Be mindful of which measurement starts as your basis (like liters or gallons) and which measurement system you need the result in.
• Watch for rounding errors, particularly in scientific and engineering contexts where precision matters.

These conversions simplify understanding measurements across different contexts and highlight practical applications, such as transferring a recipe from one measuring system to another.

Metric to Imperial Conversion

Metric to imperial conversion is a frequent necessity, as these two systems of measurement are often used in different parts of the world. It's essential to understand how to accurately make these conversions to maintain accuracy in scientific and everyday applications. To perform metric to imperial conversions for volume:

• Understand that 1 liter (L) is a volume measurement equivalent to 1000 cubic centimeters (cm³), an important part of the metric system.
• Note the conversion rate between inches and centimeters: 1 inch = 2.54 centimeters. For cubic measurements, this conversion needs to be cubed.
• When converting liters to cubic inches, you first move from liters to cubic centimeters and then from cubic centimeters to cubic inches, as shown in the exercise. This approach ensures the accuracy of your conversion.

Metric to imperial conversions can be straightforward by following a step-by-step approach, ensuring precise calculation accurately transforming units between these systems.

Volume Measurement

Volume measurement refers to the amount of space occupied by a three-dimensional object. This concept is central to many fields, including cooking, engineering, and science, where precise volume measurements are critical. Volume units come in both metric and imperial systems:

• In the metric system, the common units are the liter and cubic centimeter. The conversion between these units is straightforward: 1 liter = 1000 cm³.
• For the imperial system, cubic inches, gallons, and fluid ounces are common units.
• Volume can be calculated for regular shapes using mathematical formulas (like length x width x height for a cube) but for irregular shapes, methods like water displacement can be utilized.

Accurate volume measurement ensures correct proportions in mixtures, mandates the right capacity for containers, and supports precise scientific experimentation. Understanding and employing correct volume conversions and measurements are vital for efficiency and accuracy.