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Simplifying 2600 = 6.28r * 37 + 2 * 3.14r2 Reorder the terms for easier multiplication: 2600 = 6.28 * 37r + 2 * 3.14r2 Multiply 6.28 * 37 2600 = 232.36r + 2 * 3.14r2 Multiply 2 * 3.14 2600 = 232.36r + 6.28r2 Solving 2600 = 232.36r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2600 + -232.36r + -6.28r2 = 232.36r + -232.36r + 6.28r2 + -6.28r2 Combine like terms: 232.36r + -232.36r = 0.00 2600 + -232.36r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 2600 + -232.36r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2600 + -232.36r + -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'. -414.0127389 + 37r + r2 = 0 Move the constant term to the right: Add '414.0127389' to each side of the equation. -414.0127389 + 37r + 414.0127389 + r2 = 0 + 414.0127389 Reorder the terms: -414.0127389 + 414.0127389 + 37r + r2 = 0 + 414.0127389 Combine like terms: -414.0127389 + 414.0127389 = 0.0000000 0.0000000 + 37r + r2 = 0 + 414.0127389 37r + r2 = 0 + 414.0127389 Combine like terms: 0 + 414.0127389 = 414.0127389 37r + r2 = 414.0127389 The r term is 37r. Take half its coefficient (18.5). Square it (342.25) and add it to both sides. Add '342.25' to each side of the equation. 37r + 342.25 + r2 = 414.0127389 + 342.25 Reorder the terms: 342.25 + 37r + r2 = 414.0127389 + 342.25 Combine like terms: 414.0127389 + 342.25 = 756.2627389 342.25 + 37r + r2 = 756.2627389 Factor a perfect square on the left side: (r + 18.5)(r + 18.5) = 756.2627389 Calculate the square root of the right side: 27.500231615 Break this problem into two subproblems by setting (r + 18.5) equal to 27.500231615 and -27.500231615.Subproblem 1
r + 18.5 = 27.500231615 Simplifying r + 18.5 = 27.500231615 Reorder the terms: 18.5 + r = 27.500231615 Solving 18.5 + r = 27.500231615 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.5' to each side of the equation. 18.5 + -18.5 + r = 27.500231615 + -18.5 Combine like terms: 18.5 + -18.5 = 0.0 0.0 + r = 27.500231615 + -18.5 r = 27.500231615 + -18.5 Combine like terms: 27.500231615 + -18.5 = 9.000231615 r = 9.000231615 Simplifying r = 9.000231615Subproblem 2
r + 18.5 = -27.500231615 Simplifying r + 18.5 = -27.500231615 Reorder the terms: 18.5 + r = -27.500231615 Solving 18.5 + r = -27.500231615 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.5' to each side of the equation. 18.5 + -18.5 + r = -27.500231615 + -18.5 Combine like terms: 18.5 + -18.5 = 0.0 0.0 + r = -27.500231615 + -18.5 r = -27.500231615 + -18.5 Combine like terms: -27.500231615 + -18.5 = -46.000231615 r = -46.000231615 Simplifying r = -46.000231615Solution
The solution to the problem is based on the solutions from the subproblems. r = {9.000231615, -46.000231615}
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