Final answer:
The length of major arc xy is 18.75 units. The radius of the circle is approximately 4.77 units. The area of the shaded sector is approximately 34.91 square units.
Step-by-step explanation:
To solve this problem, we need to use the relationship between the central angle and the circumference of the circle, and then use that to find the area of the shaded sector. The length of the major arc xy can be found by taking the ratio of the central angle to the total angle in a circle (which is 360 degrees) and multiplying it by the circumference of the circle.
Since the circumference of the circle is given as 30 units and the central angle is 225°, the length of major arc xy is calculated as follows:
Length of arc xy
= (225° / 360°) × 30 units = (5/8) × 30 units = 18.75 units.
For the radius, we use the formula for the circumference of a circle, which is 2πr. Since the circumference is 30 units, we solve for r:
30 units = 2πr → r = 30 units / (2π) → r ≈ 4.77 units
Last, to find the area of the shaded sector, we take the ratio of the central angle to 360 degrees and multiply it by the area of the entire circle, πr²:
Area of shaded sector = (225° / 360°) × π × (4.77²) ² → Approx. 34.91 square units