Problem 122 - Algebra A Combined Function

Question

The formula \(S=6 x^{2}\) can be used to find the surface area \(S\) of a cube with side length \(x\). Find the surface area of a cube with side length 5 meters. (Surface area is measured in square units.)

Step-by-Step Solution

Answer

The surface area of the cube is 150 square meters.

Step 1: Identify the Variables

We are given the formula for the surface area of a cube, which is \( S = 6x^2 \), where \( S \) is the surface area and \( x \) is the side length of the cube. In this problem, we know that the side length \( x \) is 5 meters.

Step 2: Substitute into the Formula

Substitute the value of \( x \) into the surface area formula: \( S = 6(5)^2 \).

Step 3: Perform the Squaring Operation

Calculate \( (5)^2 \), which means \( 5 \times 5 = 25 \). So now our equation becomes: \( S = 6 \times 25 \).

Step 4: Multiply to Find Surface Area

Multiply the squared value by 6: \( 6 \times 25 = 150 \).

Step 5: State the Final Answer

The surface area of the cube is 150 square meters.

Key Concepts

Understanding Cube Geometry

In the world of geometry, a cube is a three-dimensional shape with six equal square faces. Each face meets another at a right angle, forming a perfectly symmetrical and uniform solid. Since all faces are square, knowing the length of one side is crucial as it determines the entire cube’s dimensions. This side length is usually denoted as \( x \).

To understand a cube’s surface area, one must visualize it. Imagine a dice used in board games; each of the six faces contributes to the total surface area. If each face is a square with sides of length \( x \), then calculating the surface area involves looking at all six square faces.

• All sides of the cube are equal.
• All angles between the faces are right angles.
• There are six faces, twelve edges, and eight vertices in a cube.

Applying the Surface Area Formula

The formula to calculate the surface area of a cube is elegant yet powerful: \( S = 6x^2 \). It stems from the cube having six identical square faces. This formula allows you to efficiently determine the surface area given the length of one side.

Acknowledging the context of surface area calculations is important. If you only know the side length \( x \), this formula applies seamlessly, providing the total area in which one might paint or cover the outside surfaces of a cube.

• Easy to remember: \( S = 6x^2 \) because six faces.
• Crucial for surface-related calculations in many practical applications.

Performing Mathematical Substitution

Mathematical substitution involves replacing variables with known values to simplify an expression or equation. In the context of our problem, we know the side length \( x \) is 5 meters. Substituting this into the formula \( S = 6x^2 \), transforms it into \( S = 6(5)^2 \).

This step is critical as it allows us to move from a general expression to a specific situation, deducing specific numeric results. Here, replacing \( x \) with the number 5 simplifies our path to the answer.

• Substitution translates abstract formulas into concrete results.
• Ensures that calculations are tailored to the problem's given values.

Executing the Squaring Operation

Squaring is a fundamental mathematical operation, particularly relevant in calculating areas, as it refers to multiplying a number by itself. When the side length in our problem is substituted as 5 meters, squaring it means computing \((5)^2 \), or \( 5 \times 5 \), which equals 25.

Understanding squaring clarifies why the unit becomes square units, which in our case, converts the meters into square meters. After performing the squaring operation, we simply multiply by 6 to finish the calculation.

• Squaring forms the basis of many area-related calculations.
• It’s essential to ensure units are appropriately squared (e.g., meters become square meters).