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Simplifying 8038.4 = 2 * 3.14r2 + 2 * 3.14r * 64 Multiply 2 * 3.14 8038.4 = 6.28r2 + 2 * 3.14r * 64 Reorder the terms for easier multiplication: 8038.4 = 6.28r2 + 2 * 3.14 * 64r Multiply 2 * 3.14 8038.4 = 6.28r2 + 6.28 * 64r Multiply 6.28 * 64 8038.4 = 6.28r2 + 401.92r Reorder the terms: 8038.4 = 401.92r + 6.28r2 Solving 8038.4 = 401.92r + 6.28r2 Solving for variable 'r'. Reorder the terms: 8038.4 + -401.92r + -6.28r2 = 401.92r + -401.92r + 6.28r2 + -6.28r2 Combine like terms: 401.92r + -401.92r = 0.00 8038.4 + -401.92r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 8038.4 + -401.92r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 8038.4 + -401.92r + -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'. -1280 + 64r + r2 = 0 Move the constant term to the right: Add '1280' to each side of the equation. -1280 + 64r + 1280 + r2 = 0 + 1280 Reorder the terms: -1280 + 1280 + 64r + r2 = 0 + 1280 Combine like terms: -1280 + 1280 = 0 0 + 64r + r2 = 0 + 1280 64r + r2 = 0 + 1280 Combine like terms: 0 + 1280 = 1280 64r + r2 = 1280 The r term is 64r. Take half its coefficient (32). Square it (1024) and add it to both sides. Add '1024' to each side of the equation. 64r + 1024 + r2 = 1280 + 1024 Reorder the terms: 1024 + 64r + r2 = 1280 + 1024 Combine like terms: 1280 + 1024 = 2304 1024 + 64r + r2 = 2304 Factor a perfect square on the left side: (r + 32)(r + 32) = 2304 Calculate the square root of the right side: 48 Break this problem into two subproblems by setting (r + 32) equal to 48 and -48.Subproblem 1
r + 32 = 48 Simplifying r + 32 = 48 Reorder the terms: 32 + r = 48 Solving 32 + r = 48 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + r = 48 + -32 Combine like terms: 32 + -32 = 0 0 + r = 48 + -32 r = 48 + -32 Combine like terms: 48 + -32 = 16 r = 16 Simplifying r = 16Subproblem 2
r + 32 = -48 Simplifying r + 32 = -48 Reorder the terms: 32 + r = -48 Solving 32 + r = -48 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + r = -48 + -32 Combine like terms: 32 + -32 = 0 0 + r = -48 + -32 r = -48 + -32 Combine like terms: -48 + -32 = -80 r = -80 Simplifying r = -80Solution
The solution to the problem is based on the solutions from the subproblems. r = {16, -80}
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