Question

A gas can holds 5.0 gal of gasoline. Express this quantity in \(\mathrm{cm}^{3}\).

Short answer

5.0 gal is equal to approximately 18,925 cm^3.

Step-by-step solution

Identify Conversion Factors

To convert from gallons to cubic centimeters, we need to know the conversion factors. There are 231 cubic inches in 1 gallon, and 2.54 centimeters in 1 inch. This will require a two-step conversion, first converting gallons to cubic inches, and then cubic inches to cubic centimeters.

Convert Gallons to Cubic Inches

Multiply the volume in gallons by the number of cubic inches in one gallon. For 5.0 gallons, this is calculated as: \(5.0 \text{ gal} \times 231 \text{ in}^{3}/\text{gal}\).

Convert Cubic Inches to Cubic Centimeters

Next, convert the volume in cubic inches to cubic centimeters. Since one cubic inch is equal to \(2.54 \text{ cm}\) per inch cubed, you need to cube 2.54 and multiply it by the volume in cubic inches. Use the formula: \( \text{volume in cm}^{3} = \text{volume in in}^{3} \times (2.54 \text{ cm/in})^{3}\).

Calculate Volume in Cubic Centimeters

Complete the calculation: \(5.0 \times 231 \times (2.54)^{3}\) to find the volume in cubic centimeters.

Simplify to Find the Final Answer

After multiplying all the values, you will get the volume of gasoline in cubic centimeters. This is the final answer to the conversion.

Key concepts

Unit Conversion

Unit conversion is a fundamental aspect of problem-solving in physics, chemistry, engineering, and everyday life. When working with different measurement systems, we often need to convert quantities from one unit to another to make appropriate comparisons or calculations.

For example, when working with volumes, you may need to convert a recipe from gallons to liters or understand how much space is taken up by a certain amount of liquid in different units. Unit conversion involves using conversion factors, which are constants that allow you to multiply or divide a quantity to change it from one unit to another. The conversion factors must represent an equivalency; that is, they must equal one when expressed as a ratio of two units. An important thing to remember is that when converting units, we are not altering the physical quantity, just expressing it in different units. Keeping track of the units and making sure they cancel out correctly is essential for the accuracy of the conversion.

To assist in these conversions, sometimes a two-step process is necessary, especially when converting between units that are not directly related, such as converting gallons to cubic centimeters, which requires an intermediate conversion to cubic inches.

Cubic Centimeters

The cubic centimeter (denoted by cm³) is a commonly used unit of volume in the metric system. One cubic centimeter is defined as the volume of a cube with sides of 1 centimeter each. This unit is especially popular in the fields of chemistry and medicine, where small volumes are frequently measured.

Understanding the scale of cubic centimeters is crucial as it represents a very small volume compared to liters or gallons often used in larger-scale measurements. For instance, there are 1,000 cubic centimeters in a liter, which demonstrates how the cubic centimeter is used to represent smaller, more precise volumes. In different contexts, the cubic centimeter might also be referred to as a milliliter (mL), which is its equivalent volume in the International System of Units (SI). Familiarity with the cubic centimeter is important for precision in scientific calculations and dimensional analysis.

Cubic Inches

A cubic inch (denoted by in³) is a unit of volume used predominantly in the United States, representing the volume of a cube with each of its sides being one inch long. It's part of the US customary units, which are different from the metric system commonly used worldwide.

In many real-world scenarios, such as in the manufacturing of engines, the cubic inch is a preferred unit for expressing the displacement of engines in cars and other vehicles. It is crucial to understand the relationship between cubic inches and other units of volume, especially when dealing with international specifications or conversions to metric units. Since the inch is a unit of length, and volume is a function of length cubed, the volume in cubic inches is found by multiplying the lengths of all three dimensions (length, width, height) if they are given in inches.

Gallons to Cubic Centimeters

Converting gallons to cubic centimeters involves understanding both of these units and how they relate within their respective systems—the US customary system for gallons and the metric system for cubic centimeters. A gallon is a larger unit for measuring volume, commonly used for liquids in the United States.

The conversion from gallons to cubic centimeters is not a direct one because it utilizes different systems of measurement. It requires a two-step conversion process: first converting the gallons to cubic inches (as one gallon is equivalent to 231 cubic inches), and then converting those cubic inches to cubic centimeters. One inch is equivalent to 2.54 centimeters; therefore, converting to cubic centimeters involves cubing 2.54 (since a volume is a three-dimensional measure), and then multiplying that by the volume in cubic inches.

Since this can be complicated, having a step-by-step approach, such as the one provided in the textbook solution, simplifies the process and minimizes errors. Understanding each step and keeping track of units throughout the conversion are essential for achieving accurate and reliable results. Comprehension of these concepts promotes better understanding of the scale of volumes used in various contexts around the world.