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Simplifying 3150.048 = 2 * 3.14r2 + 2 * 3.14r * 99.7 Multiply 2 * 3.14 3150.048 = 6.28r2 + 2 * 3.14r * 99.7 Reorder the terms for easier multiplication: 3150.048 = 6.28r2 + 2 * 3.14 * 99.7r Multiply 2 * 3.14 3150.048 = 6.28r2 + 6.28 * 99.7r Multiply 6.28 * 99.7 3150.048 = 6.28r2 + 626.116r Reorder the terms: 3150.048 = 626.116r + 6.28r2 Solving 3150.048 = 626.116r + 6.28r2 Solving for variable 'r'. Reorder the terms: 3150.048 + -626.116r + -6.28r2 = 626.116r + -626.116r + 6.28r2 + -6.28r2 Combine like terms: 626.116r + -626.116r = 0.000 3150.048 + -626.116r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 3150.048 + -626.116r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 3150.048 + -626.116r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -501.6 + 99.7r + r2 = 0 Move the constant term to the right: Add '501.6' to each side of the equation. -501.6 + 99.7r + 501.6 + r2 = 0 + 501.6 Reorder the terms: -501.6 + 501.6 + 99.7r + r2 = 0 + 501.6 Combine like terms: -501.6 + 501.6 = 0.0 0.0 + 99.7r + r2 = 0 + 501.6 99.7r + r2 = 0 + 501.6 Combine like terms: 0 + 501.6 = 501.6 99.7r + r2 = 501.6 The r term is 99.7r. Take half its coefficient (49.85). Square it (2485.0225) and add it to both sides. Add '2485.0225' to each side of the equation. 99.7r + 2485.0225 + r2 = 501.6 + 2485.0225 Reorder the terms: 2485.0225 + 99.7r + r2 = 501.6 + 2485.0225 Combine like terms: 501.6 + 2485.0225 = 2986.6225 2485.0225 + 99.7r + r2 = 2986.6225 Factor a perfect square on the left side: (r + 49.85)(r + 49.85) = 2986.6225 Calculate the square root of the right side: 54.65 Break this problem into two subproblems by setting (r + 49.85) equal to 54.65 and -54.65.Subproblem 1
r + 49.85 = 54.65 Simplifying r + 49.85 = 54.65 Reorder the terms: 49.85 + r = 54.65 Solving 49.85 + r = 54.65 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-49.85' to each side of the equation. 49.85 + -49.85 + r = 54.65 + -49.85 Combine like terms: 49.85 + -49.85 = 0.00 0.00 + r = 54.65 + -49.85 r = 54.65 + -49.85 Combine like terms: 54.65 + -49.85 = 4.8 r = 4.8 Simplifying r = 4.8Subproblem 2
r + 49.85 = -54.65 Simplifying r + 49.85 = -54.65 Reorder the terms: 49.85 + r = -54.65 Solving 49.85 + r = -54.65 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-49.85' to each side of the equation. 49.85 + -49.85 + r = -54.65 + -49.85 Combine like terms: 49.85 + -49.85 = 0.00 0.00 + r = -54.65 + -49.85 r = -54.65 + -49.85 Combine like terms: -54.65 + -49.85 = -104.5 r = -104.5 Simplifying r = -104.5Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.8, -104.5}
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