This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define the perimeter of a sector. The perimeter of a sector is the sum of the lengths of its two radii and its arc length. Let r be the radius of the circle. Let L be the arc length of the sector. The perimeter P = 2r + L. Step 2: Calculate the arc length of the sector. The sector is (2)/(3) of a circle. The circumference of a full circle is C = 2 r. The arc length L of the sector is (2)/(3) of the circumference: L = (2)/(3) (2 r) = (4 r)/(3) Step 3: Set up the equation for the perimeter. We are given that the perimeter P = 260 cm. Substitute this and the expression for L into the perimeter formula: 260 = 2r + (4 r)/(3) Step 4: Solve for the radius r. Factor out r from the right side of the equation: 260 = r (2 + (4)/(3)) Find a common denominator for the terms inside the parenthesis: 260 = r ((6)/(3) + (4)/(3)) 260 = r ((6 + 4)/(3)) Now, isolate r: r = (260 × 3)/(6 + 4) r = (780)/(6 + 4) We can factor out 2 from the denominator: r = (780)/(2(3 + 2)) r = (390)/(3 + 2) Now, substitute the approximate value of ≈ 3.14159: r = (390)/(3 + 2(3.14159)) r = (390)/(3 + 6.28318) r = (390)/(9.28318) r ≈ 41.990 Rounding to two decimal places, the radius is 41.99 cm. The radius of the sector is 41.99 cm. Send me the next one 📸