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
Okay, Samadhi, let's re-solve this using = (22)/(7) and clearly separating the arc length and radius calculations. Step 1: Define the perimeter of a sector. The perimeter P of a sector is given by the sum of its two radii and its arc length L: P = 2r + L We are given P = 260 cm. Step 2: Express the arc length L in terms of the radius r. The sector is (2)/(3) of a circle. The circumference of a full circle is C = 2 r. Therefore, the arc length L is (2)/(3) of the circumference: L = (2)/(3) (2 r) Substitute = (22)/(7): L = (2)/(3) (2 × (22)/(7) × r) L = (2)/(3) ((44)/(7) r) L = (88)/(21) r This is the expression for the arc length in terms of r. Step 3: Find the radius r. Substitute the expression for L into the perimeter formula: 260 = 2r + (88)/(21) r Factor out r: 260 = r (2 + (88)/(21)) Find a common denominator for the terms inside the parenthesis: 260 = r ((2 × 21)/(21) + (88)/(21)) 260 = r ((42)/(21) + (88)/(21)) 260 = r ((42 + 88)/(21)) 260 = r ((130)/(21)) Now, solve for r: r = (260 × 21)/(130) r = 2 × 21 r = 42 The radius of the sector is 42 cm. Step 4: Find the arc length L. Now that we have the radius r = 42 cm, we can calculate the arc length L using the formula from Step 2: L = (88)/(21) r Substitute r = 42: L = (88)/(21) × 42 L = 88 × (42)/(21) L = 88 × 2 L = 176 The arc length of the sector is 176 cm. That's 2 down. 3 left today — send the next one.