Problem 42 - Algebra 1: Concepts and Skills

Question

Find the volume of a cube when each side x is 10 feet.

Step-by-Step Solution

Answer

The volume of the cube is 1000 cubic feet.

Step 1: Define the given side length

It is given in the exercise that each side of the cube, which we call x, is 10 feet.

Step 2: Apply the formula for cube volume

The volume of a cube V is given by the formula \(V = x^3\). Substituting the given value, we get \(V = 10^3\).

Step 3: Calculate the volume

Calculate the value of \(10^3\) to get the volume. \(V = 10^3 = 1000\) cubic feet.

Key Concepts

Understanding Geometry

Geometry is a branch of mathematics focused on shapes, sizes, and the properties of space. In geometry, understanding how dimensions work is essential—it allows us to describe the size of shapes, like cubes, and understand their characteristics.

In three-dimensional geometry, we deal with solids like cubes, spheres, and pyramids. These shapes have volume, surface area, and can be defined by edges, faces, and vertices. A cube is a perfect example where all the sides (edges) are equal in length. Recognizing these dimensions is fundamental to solving problems involving space and volume.

Volume Calculation Simplified

Calculating volume is about determining how much space an object, like a cube, occupies. The volume is measured in cubic units, which makes sense because a cube has three dimensions: length, width, and height. For a cube, each of these dimensions is equal, simplifying the calculation.

The formula for calculating the volume of a cube is simply \(V = x^3\) where \(x\) is the length of one side.
This means you're multiplying the length of the side by itself, two more times.

• First multiplication gives the area of one face of the cube.
• Second multiplication expands it into the third dimension, giving its full volume.

In the example with \(x = 10\) feet, the volume calculation would be \(10 \times 10 \times 10 = 1000\) cubic feet. This calculation tells us how much space is inside the cube.

Exploring Cube Properties

A cube is a special kind of prism referred to as a regular hexahedron. All its faces are squares, and they are all equal in size.
A cube has several key properties that can be useful to know:

• It has 6 equal faces.
• All the angles in a cube are right angles.
• Every face of a cube meets another face at an edge—a total of 12 edges.
• A cube has 8 vertices (corners) where edges meet.

Knowing these properties is important when calculating not just the volume, but also the surface area or understanding symmetry in geometric contexts. The symmetry of a cube makes it a compelling subject in both practical applications and theoretical mathematics.