Question

How many \(1-\mathrm{cm}\) cubes would it take to construct a cube that is \(4 \mathrm{~cm}\) on edge?

Short answer

64 of the 1-cm cubes are needed to construct the 4-cm cube.

Step-by-step solution

Understand the Volume of a Cube

The volume of a cube is calculated using the formula V = a^3, where V is the volume and a is the length of an edge of the cube.

Calculate the Volume of the Larger Cube

For a cube that is 4 cm on each edge, use the volume formula. V = a^3 = 4^3 = 64 cubic centimeters.

Volume of a Single Small Cube

The volume of one 1-cm cube is 1^3 = 1 cubic centimeter since its sides are all 1 cm.

Determine the Number of Small Cubes Needed

Divide the volume of the large cube by the volume of a small cube to find out how many 1-cm cubes fit inside the larger cube. Number of small cubes = Volume of large cube / Volume of one small cube = 64 / 1 = 64.

Key concepts

Geometry in Chemistry

Geometry plays a critical role in understanding the spatial arrangement of atoms in molecules, which is fundamental to chemistry. Just as a cube is a three-dimensional shape composed of identical squares on each of its six faces, molecules have specific geometric shapes that determine how they interact and bond with one another.

For instance, the arrangement of atoms in a simple molecule like methane (CH4) can be thought of as analogous to a three-dimensional shape where carbon is at the center and hydrogen atoms are located at the vertices of a conceptual geometric shape known as a tetrahedron. This spatial understanding is crucial for predicting molecular properties and reactivity. In a classroom or homework setting, when students face problems involving cubic structures or other geometric concepts, they are indirectly building a foundation for understanding molecular geometry in chemistry.

Just as students calculate the volume of a cube by raising the length of an edge to the third power, chemists use analogous mathematical reasoning to predict the volume and shape of the space occupied by molecules, which is vital for interpreting and predicting chemical phenomena.

Cubic Centimeters

Cubic centimeters (cm³), often abbreviated as cc, are units of volume commonly used in the scientific and medical communities. A single cubic centimeter is equivalent to the volume of a cube with edges that are each one centimeter long.

Understanding the concept of cubic centimeters is essential in various practical situations, such as determining the displacement in engines or measuring liquids in medicine. It represents a three-dimensional space that is enclosed within the confines of a cube, much like how a shoebox has a certain volume capable of containing objects or air.

Real-life Connection

In real life, cubic centimeters can be used to measure how much space a substance occupies, such as the volume of water in a container or the size of an internal cavity within a mechanical part. In the educational context, when students learn to measure and calculate volumes in cubic centimeters, they are equipped with a valuable tool that has a broad range of applications from chemistry to everyday tasks.

Volume Calculation

Volume calculation is a mathematical process used to determine how much space a three-dimensional object occupies. For a cube, the volume is found by taking the length of one of its edges and raising it to the third power, as represented by the formula V = a³, where V denotes volume and a is the length of an edge.

Computing the volume is particularly useful in problems where you need to know how many smaller units fit inside a larger unit without any gaps or overlaps. This concept applies not only in abstract mathematical problems but also in real-life scenarios, such as packing, loading goods for shipping, or filling containers.

Link to Exercise

The exercise provided demonstrates a practical application of volume calculation. It poses the question of how many unit cubes are required to fill up a larger cube. By understanding that each small cube occupies a discrete volume of space and employing the volume formula, students can solve for the number of unit cubes needed to build the larger cube, thereby reinforcing their grasp of volume calculation in a tangible context.