Question

A cube has an edge length of 7 \(\mathrm{cm} .\) If it is divided up into \(1-\mathrm{cm}\) cubes, how many \(1-\mathrm{cm}\) cubes are there?

Short answer

343 1-cm cubes.

Step-by-step solution

Understand the Problem

We are asked to find the total number of 1-cm cubes that can fit inside a larger cube with an edge length of 7 cm. To do this, we need to calculate the volume of the large cube and then understand that the volume of each small cube is 1 cubic cm.

Calculate the Volume of the Large Cube

The volume of a cube is calculated by raising the length of one edge to the third power. In this case, the edge length is 7 cm, so the volume is given by \(7\,\text{cm} \times 7\,\text{cm} \times 7\,\text{cm}\) or \(7^3\,\text{cm}^3\).

Determine the Number of Small Cubes

Since the volume of the larger cube is \(7^3\,\text{cm}^3\) and each small cube has a volume of 1 cubic cm, we can find the total number of small cubes by dividing the larger volume by the smaller volume, which results in \(7^3 = 343\) 1-cm cubes.

Key concepts

Cube Volume

When we talk about cube volume, we're referring to the amount of space that is enclosed within a cube's six equal sides. The formula for finding the volume of a cube is quite simple—it's the length of one edge raised to the third power, represented mathematically as V = a^3, where V stands for volume and a represents the length of an edge.

To visualize this, imagine filling a cube with water—how much water you need to fill the cube completely is its volume. If a cube has an edge length of 7 centimeters, as in our exercise, we calculate its volume by multiplying 7 cm by itself three times:
Volume (V) = edge (a) × edge (a) × edge (a) = 7 cm × 7 cm × 7 cm = 343 cm³.
This tells us that the 7 cm cube has a volume of 343 cubic centimeters. Understanding cube volume is crucial for spatial reasoning and is a fundamental concept in geometry and various real-world applications, such as packaging and storage.

Unit Conversion

Unit conversion is a critical skill in mathematics, science, and many practical situations like cooking and construction. It involves changing a measure of quantity from one unit to another while maintaining the same quantity. For example, converting feet to meters or pounds to kilograms. It’s about translating different languages of measurement to one another.

In our cube exercise, we’re dealing with cubic centimeters—a unit expressing volume. We use this unit to measure the space that objects occupy. Sometimes, you might need to convert cubic centimeters to liters or cubic feet, depending on your application.
To convert units, you generally multiply by a conversion factor that represents an equivalence. For instance, to convert cm³ to liters, you might use the conversion factor 1 liter = 1000 cm³, meaning that if you had 343 cm³, it would be 0.343 liters after conversion. Understanding how to convert between different units is essential for many scientific measurements and practical tasks.

Mathematical Problem Solving

Mathematical problem solving is a process that involves understanding a problem, devising a plan to solve it, carrying out the plan, and then looking back to check and interpret the results. It’s a fundamental skill that advances logical thinking and the ability to work through complex challenges step by step.

In the context of our exercise, solving the problem begins with a clear understanding of the task—we need to find how many 1-cm³ cubes fit into a larger cube. Next, we devise a plan: calculate the volume of the large cube, and recognize that this volume represents the total number of small cubes it can contain, given that each small cube is 1 cm³. After performing the calculations, we conclude with 343 small cubes. Finally, we review our steps to ensure everything was executed correctly and the result makes sense. This four-step process is applicable to all mathematical problems and is also a valuable approach in many real-world scenarios where systematic thinking is required.