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: Understand the relationship between arc length and the circumference of a circle. The problem states that the length of an arc is (1)/(5) of its circumference. This means that the sector enclosed by this arc represents (1)/(5) of the entire circle. The ratio of the arc length to the circumference is equal to the ratio of the sector's angle to 360^. Arc LengthCircumference = Angle of Sector360^ = (1)/(5) Step 2: Understand the relationship between the area of a sector and the area of the circle. The ratio of the area of a sector to the area of the entire circle is the same as the ratio of the arc length to the circumference (or the ratio of the sector's angle to 360^). Area of SectorArea of Circle = Arc LengthCircumference Step 3: Calculate the area of the sector. Given that the area of the circle is 346.5 cm^2 and the arc length is (1)/(5) of the circumference, we can write: Area of Sector346.5 cm^2 = (1)/(5) To find the area of the sector, multiply the area of the circle by (1)/(5): Area of Sector = (1)/(5) × 346.5 cm^2 Area of Sector = 69.3 cm^2 The final answer is 69.3 cm^2. 3 done, 2 left today. You're making progress.