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 (L) = 1000 cm\(^3\) and 1 in. = 2.54 cm, express this volume in cubic inches.

Step-by-Step Solution

Answer

473 cm³ is approximately 28.86 in³.

Step 1: Convert Liters to Cubic Centimeters

Given the volume of the salad dressing is 0.473 L, we start by converting this volume into cubic centimeters. We use the conversion factor: \[1 \text{ liter} = 1000 \text{ cm}^3\].Therefore, the volume in cubic centimeters is:\[0.473 \text{ L} \times 1000 \text{ cm}^3/\text{L} = 473 \text{ cm}^3\].

Step 2: Convert Cubic Centimeters to Cubic Inches

Next, we convert the volume from cubic centimeters to cubic inches. We need to be aware of the conversion between linear centimeters and inches. Given that \[1 \text{ inch} = 2.54 \text{ cm}\], we cube both sides to find the conversion for volumes:\[1 \text{ in}^3 = (2.54 \text{ cm})^3\].This gives us:\[1 \text{ in}^3 = 16.387064 \text{ cm}^3\].Now, convert the volume from cubic centimeters to cubic inches:\[ \frac{473 \text{ cm}^3}{16.387064 \text{ cm}^3/\text{in}^3} \approx 28.86 \text{ in}^3\].

Key Concepts

Understanding Cubic Centimeters

Cubic centimeters, abbreviated as cm\(^3\) or ccs, are a unit of volume that is widely used in scientific and medical contexts. One cubic centimeter is the volume of a cube with sides that are each one centimeter long.

• A cubic centimeter is therefore quite small, roughly the size of a sugar cube.
• Due to their small size, they are often used in fields that require precise volume measurements, such as chemistry and medicine.
• One advantage of using cubic centimeters is their simplicity when converting to other metric units, like milliliters or liters.

In most cases, converting liters to cubic centimeters is straightforward. Since 1 liter equals 1,000 cubic centimeters, to find the volume in cubic centimeters, you simply multiply the number of liters by 1,000. For instance, 0.473 liters becomes straightforwardly 473 cm\(^3\). This metric system makes conversions seamless and eliminates complex calculations.

The Concept of Cubic Inches

Cubic inches are another unit of volume, typically used in everyday measurements in countries that utilize the Imperial system, such as the United States. A cubic inch is the volume of a cube with sides measuring one inch each.

• One cubic inch is larger than a cubic centimeter since one inch equals 2.54 centimeters. Consequently, a cubic inch equals about 16.387 cubic centimeters.
• This means when converting from cm\(^3\) to in\(^3\), you generally divide the number of cubic centimeters by 16.387 to achieve in\(^3\).
• This conversion is essential because devices and tools often use different volume units, depending on where they are made or used.

When converting between these units, understanding that lengths convert easier and then cube those values for volume is vital. For example, converting 473 cm\(^3\) to cubic inches involves noting that 473 cm\(^3\) divided by 16.387 results in approximately 28.86 in\(^3\). This conversion proves handy when translating between different measuring systems effectively.

Defining Liter

A liter, abbreviated as L, is a metric unit of volume commonly used for everyday liquids like milk and water. Liters offer a simple and clear way to measure volumes given they operate on a base-10 system. One liter is equivalent to the volume of a cube that is 10 centimeters on each side.

• Liters are used globally, primarily in industries, kitchens, and scientific contexts, for their ease with conversions.
• One of the most common conversions is that 1 liter equals 1,000 cubic centimeters (cm\(^3\)). This relationship stems from the metric system's design where base conversions are one thousand, making it user-friendly.
• Moreover, this unit is versatile, allowing easy conversion into milliliters (1 L = 1,000 mL) and related larger units like cubic meters (1 m\(^3\) = 1,000 L).

For example, to translate 0.473 liters into cubic centimeters, you multiply by 1,000, which results in 473 cm\(^3\), a standard conversion that simplifies many practical tasks. Understanding liters thus enables smooth transitions between different scales of volume measurement.