Substituting the values, \( V = \frac{1}{3} \pi (4)^2 (9) = \frac{1}{3} \pi \times 144 = 48\pi \approx 150.8 \) cubic centimeters.
["Understanding the Volume Calculation: Substituting Values to Find ( V = \frac{1}{3} \pi (4)^2 (9) )", "When calculating the volume of a geometric shape, selecting the correct values is crucial for accurate results. One compelling example is substituting specific dimensions into the formula ( V = \frac{1}{3} \pi r^2 h ), often used to compute the volume of a cone. Substituting ( r = 4 ) cm and ( h = 9 ) cm yields:", "[\nV = \frac{1}{3} \pi (4)^2 (9)\n]", "### Breaking Down the Formula\nThe formula ( V = \frac{1}{3} \pi r^2 h ) applies to cones — three-dimensional shapes with a circular base tapering to a point (apex). Here:\n- ( r ) is the radius of the circular base\n- ( h ) is the height (perpendicular distance from base to apex)\n- ( \pi ) (pi) is a mathematical constant approximated as 3.1416", "### Substitution Step-by-Step\nSubstituting ( r = 4 ) and ( h = 9 ):\n[\nV = \frac{1}{3} \pi (4)^2 (9)\n]", "First, calculate ( r^2 ):\n[\n(4)^2 = 16\n]", "Multiply by height:\n[\n16 \ imes 9 = 144\n]", "Now apply the ( \frac{1}{3} \pi ) factor:\n[\nV = \frac{1}{3} \pi \ imes 144 = 48\pi\n]", "### Numerical Approximation\nTo obtain a practical value, substitute ( \pi \approx 3.1416 ):\n[\n48 \ imes 3.1416 \approx 150.8 \ ext{ cubic centimeters}\n]", "### Why This Substitution Matters\nUsing accurate dimensions ensures the computed volume reflects the true storage capacity or material volume—essential in engineering, design, and manufacturing contexts. The substitution also clearly demonstrates the cone volume formula’s structure, reinforcing mathematical relationships between radius, height, and capacity.", "### Summary\nSubstituting ( r = 4 ) cm and ( h = 9 ) cm into ( V = \frac{1}{3} \pi r^2 h ) delivers a precise and meaningful result:\n[\nV = 48\pi \approx 150.8\ \ ext{cm}^3\n]\nThis precise volume calculation exemplifies how substituting correct values transforms abstract formulas into practical, measurable quantities.", "---", "Keywords: cone volume formula, calculate volume, geometry, ( \frac{1}{3} \pi r^2 h ), math example, ( \pi \approx 3.1416 ), unit conversion cm³, dimensional substitution, engineering measurement."]
