Problem 57 - Prealgebra

Question

A refrigerator has a capacity of 20 cubic feet. What is the capacity of the refrigerator in cubic inches?

Step-by-Step Solution

Answer

The refrigerator's capacity is 34,560 cubic inches.

Step 1: Understanding the Conversion Factor

Before converting, identify that 1 cubic foot equals 1,728 cubic inches, as there are 12 inches in a foot and so in volume: \[ 12 imes 12 imes 12 = 1,728 \, ext{cubic inches in a cubic foot} \]

Step 2: Calculating the Capacity in Cubic Inches

Using the conversion factor, multiply the refrigerator's cubic feet by the cubic inches per cubic foot to find the capacity in cubic inches: \[ 20 imes 1,728 = 34,560 \, ext{cubic inches} \]

Key Concepts

Cubic Feet to Cubic Inches

Volume measurement often requires converting from one unit to another, especially when dealing with different measurement systems. Cubic feet and cubic inches are both units of volume used in the imperial system. While a cubic foot measures a 3D space (a box) that is one foot in each dimension, a cubic inch measures a space that is one inch in each dimension. To convert cubic feet to cubic inches, we need to understand the basic fact that there are 12 inches in one foot.

By calculating how many inches fit into a cube where each side is one foot long, we find that these dimensions are multiplied together:

• Width: 12 inches
• Height: 12 inches
• Depth: 12 inches

Thus, the formula for conversion becomes: \[12 \times 12 \times 12 = 1,728\]
When you multiply 12 by itself three times, you get 1,728 cubic inches in one cubic foot. This is a fundamental conversion factor you will use to switch between these two units.

Capacity Calculation

Calculating the capacity of something like a refrigerator involves understanding the space inside a 3D object and how it translates between different volume units. For example, when you know the capacity of a refrigerator is 20 cubic feet, you might need to find out what that looks like in cubic inches.

Using the conversion factor discussed, the calculation becomes straightforward. You simply multiply the cubic feet by the cubic inches per cubic foot to convert the entire space. For a capacity of 20 cubic feet, it looks like this:

• First, confirm the conversion factor: 1 cubic foot = 1,728 cubic inches.
• Multiply the cubic feet by the conversion factor: \[20 \times 1,728 = 34,560\]

The refrigerator thus has a capacity of 34,560 cubic inches. Understanding how to quickly perform this calculation is essential in engineering, architecture, and other fields.

Volume Conversion Factor

A volume conversion factor is a crucial concept in unit conversion, serving as a bridge between different measurement units. In the context of converting volumes, the factor relates one unit of measurement to another on a consistent basis.

For converting cubic feet to cubic inches, the conversion factor is derived by taking the linear conversion factor between feet and inches (12 inches equals 1 foot) and applying it to each dimension of volume. Since the dimensions of volume are raised to the third power (for 3D space), the conversion factor becomes:

• \[12^3 = 1,728\]
• This means 1 cubic foot = 1,728 cubic inches.

Using the volume conversion factor ensures that you consistently and accurately convert cubic measurements, no matter which directions the measurements need to go. Whether working with refrigerators or rooms, having the conversion factor readily available allows for seamless calculations.