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Simplifying 11124.393 = 2 * 3.14r2 + 2 * 3.14r * 82.1 Multiply 2 * 3.14 11124.393 = 6.28r2 + 2 * 3.14r * 82.1 Reorder the terms for easier multiplication: 11124.393 = 6.28r2 + 2 * 3.14 * 82.1r Multiply 2 * 3.14 11124.393 = 6.28r2 + 6.28 * 82.1r Multiply 6.28 * 82.1 11124.393 = 6.28r2 + 515.588r Reorder the terms: 11124.393 = 515.588r + 6.28r2 Solving 11124.393 = 515.588r + 6.28r2 Solving for variable 'r'. Reorder the terms: 11124.393 + -515.588r + -6.28r2 = 515.588r + -515.588r + 6.28r2 + -6.28r2 Combine like terms: 515.588r + -515.588r = 0.000 11124.393 + -515.588r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 11124.393 + -515.588r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 11124.393 + -515.588r + -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'. -1771.400159 + 82.1r + r2 = 0 Move the constant term to the right: Add '1771.400159' to each side of the equation. -1771.400159 + 82.1r + 1771.400159 + r2 = 0 + 1771.400159 Reorder the terms: -1771.400159 + 1771.400159 + 82.1r + r2 = 0 + 1771.400159 Combine like terms: -1771.400159 + 1771.400159 = 0.000000 0.000000 + 82.1r + r2 = 0 + 1771.400159 82.1r + r2 = 0 + 1771.400159 Combine like terms: 0 + 1771.400159 = 1771.400159 82.1r + r2 = 1771.400159 The r term is 82.1r. Take half its coefficient (41.05). Square it (1685.1025) and add it to both sides. Add '1685.1025' to each side of the equation. 82.1r + 1685.1025 + r2 = 1771.400159 + 1685.1025 Reorder the terms: 1685.1025 + 82.1r + r2 = 1771.400159 + 1685.1025 Combine like terms: 1771.400159 + 1685.1025 = 3456.502659 1685.1025 + 82.1r + r2 = 3456.502659 Factor a perfect square on the left side: (r + 41.05)(r + 41.05) = 3456.502659 Calculate the square root of the right side: 58.792028873 Break this problem into two subproblems by setting (r + 41.05) equal to 58.792028873 and -58.792028873.Subproblem 1
r + 41.05 = 58.792028873 Simplifying r + 41.05 = 58.792028873 Reorder the terms: 41.05 + r = 58.792028873 Solving 41.05 + r = 58.792028873 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-41.05' to each side of the equation. 41.05 + -41.05 + r = 58.792028873 + -41.05 Combine like terms: 41.05 + -41.05 = 0.00 0.00 + r = 58.792028873 + -41.05 r = 58.792028873 + -41.05 Combine like terms: 58.792028873 + -41.05 = 17.742028873 r = 17.742028873 Simplifying r = 17.742028873Subproblem 2
r + 41.05 = -58.792028873 Simplifying r + 41.05 = -58.792028873 Reorder the terms: 41.05 + r = -58.792028873 Solving 41.05 + r = -58.792028873 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-41.05' to each side of the equation. 41.05 + -41.05 + r = -58.792028873 + -41.05 Combine like terms: 41.05 + -41.05 = 0.00 0.00 + r = -58.792028873 + -41.05 r = -58.792028873 + -41.05 Combine like terms: -58.792028873 + -41.05 = -99.842028873 r = -99.842028873 Simplifying r = -99.842028873Solution
The solution to the problem is based on the solutions from the subproblems. r = {17.742028873, -99.842028873}
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