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Simplifying 1909 = 2 * 3.14r2 + 2 * 3.14r * 30 Multiply 2 * 3.14 1909 = 6.28r2 + 2 * 3.14r * 30 Reorder the terms for easier multiplication: 1909 = 6.28r2 + 2 * 3.14 * 30r Multiply 2 * 3.14 1909 = 6.28r2 + 6.28 * 30r Multiply 6.28 * 30 1909 = 6.28r2 + 188.4r Reorder the terms: 1909 = 188.4r + 6.28r2 Solving 1909 = 188.4r + 6.28r2 Solving for variable 'r'. Reorder the terms: 1909 + -188.4r + -6.28r2 = 188.4r + -188.4r + 6.28r2 + -6.28r2 Combine like terms: 188.4r + -188.4r = 0.0 1909 + -188.4r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 1909 + -188.4r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 1909 + -188.4r + -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'. -303.9808917 + 30r + r2 = 0 Move the constant term to the right: Add '303.9808917' to each side of the equation. -303.9808917 + 30r + 303.9808917 + r2 = 0 + 303.9808917 Reorder the terms: -303.9808917 + 303.9808917 + 30r + r2 = 0 + 303.9808917 Combine like terms: -303.9808917 + 303.9808917 = 0.0000000 0.0000000 + 30r + r2 = 0 + 303.9808917 30r + r2 = 0 + 303.9808917 Combine like terms: 0 + 303.9808917 = 303.9808917 30r + r2 = 303.9808917 The r term is 30r. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30r + 225 + r2 = 303.9808917 + 225 Reorder the terms: 225 + 30r + r2 = 303.9808917 + 225 Combine like terms: 303.9808917 + 225 = 528.9808917 225 + 30r + r2 = 528.9808917 Factor a perfect square on the left side: (r + 15)(r + 15) = 528.9808917 Calculate the square root of the right side: 22.999584598 Break this problem into two subproblems by setting (r + 15) equal to 22.999584598 and -22.999584598.Subproblem 1
r + 15 = 22.999584598 Simplifying r + 15 = 22.999584598 Reorder the terms: 15 + r = 22.999584598 Solving 15 + r = 22.999584598 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + r = 22.999584598 + -15 Combine like terms: 15 + -15 = 0 0 + r = 22.999584598 + -15 r = 22.999584598 + -15 Combine like terms: 22.999584598 + -15 = 7.999584598 r = 7.999584598 Simplifying r = 7.999584598Subproblem 2
r + 15 = -22.999584598 Simplifying r + 15 = -22.999584598 Reorder the terms: 15 + r = -22.999584598 Solving 15 + r = -22.999584598 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + r = -22.999584598 + -15 Combine like terms: 15 + -15 = 0 0 + r = -22.999584598 + -15 r = -22.999584598 + -15 Combine like terms: -22.999584598 + -15 = -37.999584598 r = -37.999584598 Simplifying r = -37.999584598Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.999584598, -37.999584598}
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