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Chapter 1: Problem 22

How many cubic inches are there in a cubic foot? The answer is not 12 .(answer check available at lightandmatter.com)

Short Answer

Expert verified
There are 1728 cubic inches in a cubic foot.

Step by step solution

01

Understanding the Problem

We need to find out how many cubic inches are in a cubic foot. Since both inches and feet are units of length, a cubic foot means a cube where each side is one foot long, and we need to convert it into cubic inches.
02

Convert Feet to Inches

First, understand that 1 foot is equal to 12 inches. This means that each side of our cube, originally 1 foot, is now 12 inches long.
03

Calculate the Volume in Cubic Inches

Since a cubic foot is a cube with each side of 12 inches, we calculate the volume by using the formula for the volume of a cube, which is side length cubed: \[ V = s^3 \] Given that the side length \( s \) is 12 inches, the volume will be: \[ V = 12^3 \]
04

Compute the Cubic Inches

Calculate \( 12^3 \): \[ 12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728 \] Therefore, there are 1728 cubic inches in a cubic foot.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Calculation
Volume calculation is a fundamental concept in geometry that deals with finding the amount of space occupied by a three-dimensional object. For cubes, calculating volume is straightforward because all sides are equal in length. The key formula to remember is:
\[ V = s^3 \]
Where \( V \) represents volume, and \( s \) is the length of a side.
  • For example, if a cube has a side length of 3 units, its volume would be \( 3^3 = 27 \) cubic units.
  • In our problem, each side of the cube is 12 inches after conversion from feet, which means the volume is \( 12^3 = 1728 \) cubic inches.
Calculating volume accurately is important because it helps us understand the capacity or space an object can hold. In this exercise, understanding the conversion and calculation led us to the realization that a cubic foot contains 1728 cubic inches.
Unit Conversion
Unit conversion is crucial when measuring different quantities to make sure everything is consistent and correct. It involves converting a measurement from one unit to another. This ensures that we can accurately calculate and compare different values.
In this exercise, converting units from feet to inches was essential:
  • 1 foot is equivalent to 12 inches.
  • For a cubic measurement, this conversion needs to be applied to length, width, and height individually, and then used in calculating volume.
Converting units correctly lets us ensure all dimensions align perfectly. It avoids confusion and errors in further calculations. Whether working with length, area, or volume, understanding unit conversion is key to accurate problem-solving.
Cubic Measurements
Cubic measurements provide a way to understand the volume of space a three-dimensional object occupies. These measurements are expressed in cubic units, which are the result of multiplying a measurement in three dimensions: length, width, and height.
  • For instance, a cube 1 foot on each side is not just 12 inches long on each side; it transforms into a cubic measurement because we consider the volume.
  • Using the formula \( V = s^3 \), we find that a cubic foot equals 1728 cubic inches due to this multiplication across three dimensions \( 12 \times 12 \times 12 \).
Understanding cubic measurements is about recognizing this three-dimensional space, unlike linear or square measurements. These measurements are crucial in many everyday calculations, from storage space to shipping container sizes, helping us quantify how much space an object truly occupies.

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