The area is \(w \times 3w = 8 \times 24 = 192\).
["Solving the Equation: Area Calculation of a Rectangle with Dimensions ( w \ imes 3w )", "Understanding how to calculate the area of a rectangle is fundamental in geometry, and one common problem involves variables in the dimensions. In this article, we explore the equation ( w \ imes 3w = 8 \ imes 24 = 192 ), explaining the steps to solve for the unknown width ( w ), interpret the result, and apply this concept to real-world geometry problems.", "---", "### Understanding Rectangle Area", "The area of a rectangle is calculated using the formula:", "[\n\ ext{Area} = \ ext{length} \ imes \ ext{width}\n]", "If one dimension is expressed as ( w ) and the other as ( 3w ), substituting into the formula gives:", "[\n\ ext{Area} = w \ imes 3w = 3w^2\n]", "---", "### Applying the Given Equation", "You’ve seen the equation:", "[\nw \ imes 3w = 8 \ imes 24 = 192\n]", "This demonstrates how to compute the product:", "1. Multiply ( w ) by ( 3w ), resulting in ( 3w^2 ).\n2. Recognize that ( 8 \ imes 24 = 192 ), which equals the area expression ( 3w^2 ).", "Thus:", "[\n3w^2 = 192\n]", "---", "### Solving for ( w )", "To find ( w ), follow these algebraic steps:", "1. Divide both sides by 3:", "[\nw^2 = \frac{192}{3} = 64\n]", "2. Take the square root of both sides:", "[\nw = \sqrt{64} = 8\n]", "So, the width ( w ) is 8 units.", "---", "### Find the Full Dimensions", "Now that ( w = 8 ), compute the full length:", "[\n3w = 3 \ imes 8 = 24\n]", "The rectangle has:", "- Width = ( 8 )\n- Length = ( 24 )\n- Area = ( 8 \ imes 24 = 192 )", "---", "### Why This Matters", "Calculating areas using algebraic expressions helps solve real-world problems such as:", "- Designing rooms or plots with variable dimensions\n- Computing surface areas in architecture and engineering\n- Solving optimization and scaling questions in math competitions", "Understanding these foundational steps builds confidence in working with equations involving geometric expressions.", "---", "### Summary", "Given a rectangle with dimensions ( w \ imes 3w ) and an area expressed as ( 8 \ imes 24 = 192 ):", "- Area = ( 3w^2 = 192 )\n- Solve: ( w^2 = 64 ) → ( w = 8 )\n- Dimensions: width = 8, length = 24, area = 192", "This method is essential for mastering algebraic geometry and stands as a key formula in solving variable-based area problems.", "---", "Keywords: rectangle area calculation, algebra equation solving, geometric formulas, variable width rectangle, area = 3w², math problem solving, geometry basics, linear equations, algebra 1, solving for w, rectangle dimensions, mathematical problem solving\nMeta Description: Learn how to solve for ( w ) in the rectangle area equation ( w \ imes 3w = 8 \ imes 24 = 192 ); step-by-step explanation and real-world application."]
