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Simplifying 3668.8388 = 2 * 3.14r(r) + 2 * 3.14r * 40.4 Multiply 2 * 3.14 3668.8388 = 6.28r * r + 2 * 3.14r * 40.4 Multiply r * r 3668.8388 = 6.28r2 + 2 * 3.14r * 40.4 Reorder the terms for easier multiplication: 3668.8388 = 6.28r2 + 2 * 3.14 * 40.4r Multiply 2 * 3.14 3668.8388 = 6.28r2 + 6.28 * 40.4r Multiply 6.28 * 40.4 3668.8388 = 6.28r2 + 253.712r Reorder the terms: 3668.8388 = 253.712r + 6.28r2 Solving 3668.8388 = 253.712r + 6.28r2 Solving for variable 'r'. Reorder the terms: 3668.8388 + -253.712r + -6.28r2 = 253.712r + -253.712r + 6.28r2 + -6.28r2 Combine like terms: 253.712r + -253.712r = 0.000 3668.8388 + -253.712r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 3668.8388 + -253.712r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 3668.8388 + -253.712r + -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'. -584.21 + 40.4r + r2 = 0 Move the constant term to the right: Add '584.21' to each side of the equation. -584.21 + 40.4r + 584.21 + r2 = 0 + 584.21 Reorder the terms: -584.21 + 584.21 + 40.4r + r2 = 0 + 584.21 Combine like terms: -584.21 + 584.21 = 0.00 0.00 + 40.4r + r2 = 0 + 584.21 40.4r + r2 = 0 + 584.21 Combine like terms: 0 + 584.21 = 584.21 40.4r + r2 = 584.21 The r term is 40.4r. Take half its coefficient (20.2). Square it (408.04) and add it to both sides. Add '408.04' to each side of the equation. 40.4r + 408.04 + r2 = 584.21 + 408.04 Reorder the terms: 408.04 + 40.4r + r2 = 584.21 + 408.04 Combine like terms: 584.21 + 408.04 = 992.25 408.04 + 40.4r + r2 = 992.25 Factor a perfect square on the left side: (r + 20.2)(r + 20.2) = 992.25 Calculate the square root of the right side: 31.5 Break this problem into two subproblems by setting (r + 20.2) equal to 31.5 and -31.5.Subproblem 1
r + 20.2 = 31.5 Simplifying r + 20.2 = 31.5 Reorder the terms: 20.2 + r = 31.5 Solving 20.2 + r = 31.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-20.2' to each side of the equation. 20.2 + -20.2 + r = 31.5 + -20.2 Combine like terms: 20.2 + -20.2 = 0.0 0.0 + r = 31.5 + -20.2 r = 31.5 + -20.2 Combine like terms: 31.5 + -20.2 = 11.3 r = 11.3 Simplifying r = 11.3Subproblem 2
r + 20.2 = -31.5 Simplifying r + 20.2 = -31.5 Reorder the terms: 20.2 + r = -31.5 Solving 20.2 + r = -31.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-20.2' to each side of the equation. 20.2 + -20.2 + r = -31.5 + -20.2 Combine like terms: 20.2 + -20.2 = 0.0 0.0 + r = -31.5 + -20.2 r = -31.5 + -20.2 Combine like terms: -31.5 + -20.2 = -51.7 r = -51.7 Simplifying r = -51.7Solution
The solution to the problem is based on the solutions from the subproblems. r = {11.3, -51.7}
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