Given \( C = 31.4 \), we have \( 2 imes 3.14 imes r = 31.4 \).
["# Solving for ( r ) in the Equation: ( 2 \ imes 3.14 \ imes r = 31.4 )", "When presented with the equation\n[\n2 \ imes 3.14 \ imes r = 31.4,\n]\nthe goal is to isolate and solve for ( r ). This type of algebraic problem is common in geometry, physics, and everyday calculations involving area or rates. In this case, the structure of the equation reveals a practical application involving circular measurements, where 3.14 likely represents an approximation of ( \pi ).", "## Understanding the Equation Structure", "Start with the given:\n[\n2 \ imes 3.14 \ imes r = 31.4\n]\nMultiplying constants on the left gives:\n[\n6.28r = 31.4\n]\nNow, solving for ( r ) involves dividing both sides by 6.28:\n[\nr = \frac{31.4}{6.28}\n]", "## Simplifying the Division", "Divide 31.4 by 6.28:\n[\nr = 5\n]", "This result, ( r = 5 ), is straightforward, but interpreting the role of ( \pi ) here is meaningful. Since ( \pi \approx 3.14 ), the equation originally incorporates ( \pi ) to reflect a circular measurement—such as circumference calculations.", "Rendering the full original expression:\n[\n2 \ imes \pi \ imes r = 31.4\n]\nSubstituting ( \pi \approx 3.14 ) and ( 2 \ imes 3.14 = 6.28 ), the equation reduces cleanly to:\n[\n6.28r = 31.4\n]\nA clear algebraic path emerges to isolate ( r ), demonstrating how real-world constants like ( \pi ) integrate into everyday math problems.", "## Why This Equation Matters", "This formula has practical applications in geometry and physics. For instance, if ( r ) represents the radius of a circle, then:\n[\n\ ext{Circumference} = 2\pi r = 31.4\n]\nGiven ( 2\pi r = 31.4 ), dividing by ( 2\pi ) yields ( r = 5 ), which can be verified:\n[\n2 \ imes \pi \ imes 5 = 10\pi \approx 10 \ imes 3.14 = 31.4\n]\nThus, the equation validates that with radius ( 5 ) and ( \pi \approx 3.14 ), the circumference measures exactly 31.4 units.", "## Conclusion", "The equation ( 2 \ imes 3.14 \ imes r = 31.4 ), large and simple as it appears, exemplifies how mathematical principles underpin real-world measurements. By recognizing ( 3.14 ) as ( \pi ), and applying basic algebra, we find:\n[\nr = 5\n]\nThis result not only solves the equation but also confirms a fundamental geometric truth: a circle with circumference 31.4 and radius multiplied by ( 2\pi ) yields a precise, manageable value.", "For learners studying geometry, physics, or engineering, mastering such equations strengthens your ability to connect theory with application—one disciplined step at a time.", "---", "Keywords: solve for ( r ), equation ( 2 \ imes 3.14 \ imes r = 31.4 ), solve linear equation, circumference formula, ( \pi \approx 3.14 ), algebraic problem solving, geometry application."]
