Problem 104 - Elementary Algebra

Question

Calculate the volume of a cube with sides measuring 3 centimeters. \((V=s 3)\)

Step-by-Step Solution

Answer

The volume of the cube is 27 cubic centimeters.

Step 1: Understanding the Problem

We are tasked with finding the volume of a cube. The formula to find the volume of a cube is given by \( V = s^3 \), where \( s \) is the length of a side.

Step 2: Identifying the Side Length

According to the problem, each side of the cube measures 3 centimeters. Thus, \( s = 3 \text{ cm} \).

Step 3: Substitute and Calculate

Substitute \( s = 3 \) into the volume formula: \[V = 3^3\]

Step 4: Calculating the Volume

Calculate the power: \[3^3 = 3 \times 3 \times 3 = 27\]Thus, the volume \( V \) is 27 cubic centimeters.

Key Concepts

cube

A cube is a three-dimensional shape that has equal length, width, and height. All the sides of a cube are squares. This makes cube a special case of a prism, where each face is not only equal in size, but also perfectly aligned at right angles. Cubes are fascinating because of their symmetry and simplicity, making them an easier shape to understand in geometry.

• All six faces of a cube are identical squares.
• Cubes have 12 equal edges.
• Cubes have 8 vertices where the corners meet.

Simple things like dice, ice cubes, and some boxes are commonly shaped as cubes in real life!

geometric formulas

Geometric formulas are essential mathematical equations used to calculate different properties of geometric figures. They help us figure out aspects like area, volume, perimeter, and more. Understanding the right formula for a shape is crucial in solving geometry problems correctly.
For a cube, these formulas revolve around its side length. The formula for the volume of a cube is one such formula using the side length:

• Volume: \( V = s^3 \), here \( s \) is the side length.

Such formulas allow us to derive accurate measurements crucial in both academic and practical scenarios.

volume calculation

Volume calculation is the process of determining the amount of space inside a three-dimensional shape. For solid objects like a cube, volume is measured in cubic units. The calculation becomes very straightforward once we know the formula and have the required measurements.
In the case of a cube:

• Measure the side length: Here it is 3 centimeters.
• Apply the formula \( V = s^3 \): Substitute \( s = 3 \)
• Calculate: \( V = 3^3 = 27 \)

Thus, the cube's volume is 27 cubic centimeters, showing how a geometric formula brings simplicity to volume calculation.

power of numbers

The power of numbers, also known as exponents, helps us express repeated multiplication in a compact form. Understanding this concept is crucial when dealing with volume, especially for cubes. In our exercise, the cube of a number \( s^3 \) is simply the number multiplied by itself two more times.

• For instance, \( 3^3 = 3 \times 3 \times 3 \).
• This gives a product of 27, representing the volume in cubic centimeters.

This highlights how exponents simplify calculations by reducing long multiplications into manageable numbers, crucial for solving problems involving geometric shapes.