Problem 50 - Algebra 2

Question

REVIEW If the surface area of a cube is increased by a factor of \(9,\) how is the length of the side of the cube changed? \(\mathbf{F}\) It is 2 times the original length. \(\mathbf{G}\) It is 3 times the original length. \(\mathbf{H}\) It is 4 times the original length. \(\mathbf{J}\) It is 5 times the original length.

Step-by-Step Solution

Answer

The side length is 3 times the original length, answer is G.

Step 1: Understand the Problem

We are given that the surface area of a cube increases by a factor of 9. We need to determine the change in the side length of the cube.

Step 2: Recall the Formula for Cube Surface Area

The surface area of a cube with side length \(s\) is calculated as \(6s^2\).

Step 3: Set Up the Equation for Increased Surface Area

Let the original side length be \(s\). If the surface area is increased by a factor of 9, the new surface area is \(9 \times 6s^2 = 54s^2\).

Step 4: Express New Surface Area in Terms of New Side Length

Let the new side length be \(s'\). The new surface area will also be \(6(s')^2\). Thus, we have the equation: \[ 6(s')^2 = 54s^2 \]

Step 5: Solve for New Side Length \(s'\)

Divide both sides of the equation by 6 to simplify: \[ (s')^2 = 9s^2 \] Take the square root of both sides: \[ s' = 3s \]

Step 6: Conclusion

The new side length \(s'\) is 3 times the original side length \(s\).

Key Concepts

Geometric Transformations

Understanding geometric transformations is vital in math, especially when dealing with shapes like cubes. They involve changing a shape's position or size while maintaining its form. In the context of surface area adjustments, this change often reflects in the dimensions of the shape.
When discussing the exercise of altering a cube, we're focusing on an enlargement, a type of geometric transformation. Here, enlarging the surface area means increasing the size of the whole cube.

• Scale Factor: This is essential in enlargements. If a geometric shape is transformed (like resizing), all sides change by a consistent factor.
• Impact on Dimensions: In a cube, if the surface area is increased by a factor of 9, similar transformations also increase the dimensions.
• Understanding Proportional Changes: Changes in surface areas will directly affect side lengths, emphasizing the consistent scaling across all dimensions.

The goal of the exercise is to determine the change in cube dimensions when experiencing such transformations. By identifying the scale factor needed for the area, one can deduce the change in side lengths, as shown in the solved exercise.

Cube Properties

Cubes are unique geometric shapes, possessing equal sides and equal angles, resulting in intriguing properties. Understanding these characteristics can make mathematical problem-solving more intuitive.
Each cube has several key properties:

• Six Faces: Equally sized squares form the surface of the cube. Hence, the total surface area of a cube with side length \(s\) is \(6s^2\).
• Uniform Edges: All edges in a cube are the same length. This property simplifies calculations when dealing with volume or surface area.
• Symmetrical Structure: Due to its uniformity, mathematical calculations involving cubes often lead to straightforward, symmetrical equations.
• Interconnected Dimensions: Any change in one dimension affects the others, maintaining their equal status.

These properties come into play when determining changes in dimension, as seen when the area's transformation leads to a change in side lengths.

Mathematical Problem Solving

Mathematical problem solving requires a strategic approach, especially with transformations and properties of cubes. The exercise involved different problem-solving techniques such as:

• Understanding the Problem: Begin by breaking down the problem, identifying what is known and what needs to be found.
• Using Relevant Formulas: Recall important formulas—like the surface area of a cube—to set up equations. The exercise uses \(6s^2\) for the original surface area of a cube.
• Setting Up Equations: Translate the problem into mathematical terms, which involves forming accurate equations as derived from known information.
• Simplifying and Solving: When you have an equation, like \(6(s')^2 = 54s^2\) from the exercise, simplify it to solve for unknowns.
• Reviewing Solution: Always verify by backtracking through steps, ensuring each part makes logical sense and that the conversion maintains integrity.

Approaching problems with this structure not only aids in finding solutions but also enhances understanding, as demonstrated by the step-by-step approach in the solved exercise.