Surface area: \(2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94\).
["Understanding Surface Area with the Equation: (2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94)", "When studying geometry and algebra, surface area calculations often involve formulas derived from three-dimensional shapes, particularly rectangular prisms. One intriguing equation that models such a scenario is:", "[\n2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94\n]", "This equation represents the total surface area of a rectangular prism (also known as a rectangular box), where (x) stands for one side length, and the dimensions (2x), (3x) represent the other length and height.", "---", "### Step-by-Step Explanation of the Surface Area Formula", "The surface area of a rectangular prism is calculated using the formula:", "[\n\ ext{Surface Area} = 2(lw + lh + wh)\n]", "In the given equation, the terms inside the parentheses correspond to the area of each pair of opposite faces:", "- (x \cdot 2x) corresponds to the width and length ((2x = 2 \cdot x)),\n- (2x \cdot 3x) corresponds to length and height ((3x = 3 \cdot x)),\n- (3x \cdot x) corresponds to width and height ((x = x)).", "Multiplying each area by 2 accounts for the two identical faces along each pair.", "---", "### Expanding the Equation", "Let’s expand the left side of the equation step-by-step:", "[\n2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 2(2x^2 + 6x^2 + 3x^2)\n]", "Combine like terms inside the parentheses:", "[\n2x^2 + 6x^2 + 3x^2 = 11x^2\n]", "Now multiply by 2:", "[\n2(11x^2) = 22x^2\n]", "So the equation simplifies to:", "[\n22x^2 = 94\n]", "---", "### Solving for (x)", "To find (x), divide both sides by 22:", "[\nx^2 = \frac{94}{22} = \frac{47}{11}\n]", "Take the square root:", "[\nx = \sqrt{\frac{47}{11}}\n]", "---", "### Interpretation and Real-World Application", "This value of (x) represents the length of a linear dimension in a rectangular prism whose total surface area is 94 square units. The formula exemplifies how algebraic expressions model real-world geometry. By solving surface area equations like this, students and engineers alike can determine physical sizes approved for packaging, construction, or manufacturing.", "---", "### Key Takeaways", "- The surface area formula uses (2(lw + lh + wh)) to sum areas of all faces.\n- Expanding and simplifying such expressions clarifies relationships between dimensions and total surface area.\n- Solving equations involving surface area helps determine missing dimensions when total area is known.\n- The equation (2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94) models a standard box-like prism with proportional side ratios.", "---", "#### Final Notes", "If you're studying geometry, mastering surface area equations like this one strengthens your analytical skills in both mathematics and applied sciences. Combining algebra with spatial reasoning enables clearer problem-solving and real-world application—key aspects for students, educators, and professionals alike.", "---", "Keywords: Surface area, 2(x·2x + 2x·3x + 3x·x) = 94, rectangular prism, geometry, algebra, solving equations, formulas, 3D shapes, math education", "Meta Description:\nExplore the surface area equation (2(x \cdot 2x + 2x \cdot 3x + 3x \cdot x) = 94), learn how to expand, simplify, and solve it, and understand its real-world geometric meaning in prisms and boxes. Ideal for students and educators."]
