Problem 145 - Chemistry A Molecular Approach

Question

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

Step-by-Step Solution

Answer

The cube with an edge length of 7 cm has a volume of 343 cubic cm, and since the volume of a 1-cm cube is 1 cubic cm, there are 343 1-cm cubes in the larger cube.

Step 1: Understanding the problem

We are given a cube with an edge length of 7 cm and we need to calculate how many 1-cm cubes can fit within this larger cube.

Step 2: Calculating the volume of the large cube

To find out how many 1-cm cubes fit inside the larger cube, first calculate the volume of the large cube using the formula for the volume of a cube, which is edge length cubed: Volume = edge length^3.

Step 3: Calculating the volume of a 1-cm cube

The volume of a 1-cm cube is 1 cm x 1 cm x 1 cm, which equals 1 cubic centimeter.

Step 4: Dividing the volumes

To find the number of 1-cm cubes, divide the volume of the larger cube by the volume of a 1-cm cube. This will give us the number of smaller cubes that fit into the larger cube.

Key Concepts

Geometry

Geometry, one of the classical disciplines of mathematics, deals with various shapes and figures, including properties, measurements, and space. One of the most fundamental three-dimensional shapes in geometry is the cube. A cube is a special kind of rectangular solid that has six equal square faces, and all its edges are the same length. It's a perfect example to understand geometry because of its symmetry and uniformity.

For anyone looking to comprehend geometry, starting with simple shapes like cubes can build a strong foundation. Grasping how to approach three-dimensional shapes is crucial because it sets the stage for understanding more complex figures in geometric space. Each of the six faces of the cube is a square, and every angle is a right angle, making it an ideal shape for introducing the concepts of volume and area in a geometrical context.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that involves understanding a problem, devising a plan to solve it, and then carrying out that plan. This process is beautifully illustrated in the cube exercise. In our cube exercise, the first step of problem solving is to understand what's being asked - how many smaller cubes of a known volume fit into a larger cube?

Understanding the dimensions given and visualizing the situation is crucial. After grasping the problem, the next step involves applying mathematical formulas—that is, using the formula for calculating the volume of a cube. Problem-solving in mathematics doesn't just involve doing calculations; it's also about logical reasoning and making connections between different concepts, such as volume and division in this scenario. By dividing the volume of the larger cube by the volume of the smaller cube, we apply a mathematical method to arrive at the solution.

Volume Calculation

Volume calculation is a part of mathematics that deals with determining the amount of space that a three-dimensional object occupies. It's a pivotal concept in multiple fields, not just in geometry class but in real-world applications like packing, construction, and manufacturing. In the context of our exercise, volume calculation allows us to figure out how many smaller units fit into a larger one, based on their sizes.

To calculate the volume of a cube, the formula is simple: \( \text{Volume} = \text{edge length}^3 \). So, for a cube with an edge length of 7 cm, the volume would be \( 7 \text{ cm} \times 7 \text{ cm} \times 7 \text{ cm} = 343 \text{ cm}^3 \). This step-by-step approach breaks down the volume calculation into an understandable process. When we know the volume of both the larger cube and the smaller 1-cm cube (\( 1 \text{ cm}^3 \)), we can find how many smaller cubes fit by performing a division, answering the practical question posed by the exercise.