The area is given by \( w \times 3w = 192 \).
["Finding the Value of ( w ) in the Area Equation: ( w \ imes 3w = 192 )", "When solving geometry or algebra problems involving area, the equation ( w \ imes 3w = 192 ) frequently appears. If you’re wondering how to find the dimension ( w ) that satisfies this area expression, you're in the right place!", "### Understanding the Equation", "The area is defined by multiplying width ( w ) by three times the same width:\n[\nw \ imes 3w = 192\n]", "This simplifies to:\n[\n3w^2 = 192\n]", "### Step-by-Step Solution", "1. Divide both sides by 3 to isolate ( w^2 ):\n[\nw^2 = \frac{192}{3} = 64\n]", "2. Take the square root of both sides to solve for ( w ):\n[\nw = \sqrt{64} = 8\n]", "Since width cannot be negative in real-world contexts, we discard the negative root.", "### Final Answer", "[\nw = 8\n]", "### Why This Matters", "Knowing ( w = 8 ) allows you to determine the full dimensions of a shape—such as a rectangular garden or panel—where width is a key variable. Given ( w = 8 ) and ( \ ext{length} = 3w = 24 ), the area is confirmed:\n[\n8 \ imes 24 = 192 \quad \checkmark\n]", "### Use in Real-Life Applications", "- Construction & Design: Calculating wall dimensions or material needs\n- Gardening: Planning plots with proportional spacing\n- Engineering: Sizing components based on area constraints", "### Summary", "To solve ( w \ imes 3w = 192 ), simplify to ( 3w^2 = 192 ) and solve for ( w ):\n[\nw = 8\n]\nThen, the corresponding length is ( 3w = 24 ), confirming the area matches. This algebraic method is essential for solving real-world area problems efficiently.", "---", "Related Keywords:\nSolve for ( w ), area formula algebra, rectangular area problem, ( w \ imes 3w = 192 \ solution, mathematics tutorial, solve quadratic equations, geometry problems, algebra equation simplification."]
