Final answer:
The area of the sector associated with arc XY, given a circle with a radius of 24 units and a central angle of 75 degrees, can be computed using the formula for the sector area, which is (75/360) × π × (24)², resulting in the area in square units.
Step-by-step explanation:
The question pertains to calculating the area of a sector in a circle given the radius and the central angle in degrees. The radius of circle O is given as 24 units, and the central angle for arc XY is given as 75 degrees. To find the area of the sector associated with arc XY, we use the formula for the area of a sector which is (θ/360) × πr², where θ is the central angle in degrees and r is the radius of the circle.
For our specific question: Area of the sector = (75/360) × π × (24)². When we calculate this, we get the area of the sector to be 1/4.8 of the area of the whole circle, which simplifies to 1/4.8 × π × 576. Multiplying these numbers together will give the final area in square units.
Step-by-step calculation
Write down the formula for the area of a sector: (θ/360) × πr².
- Insert the given values into the formula: (75 degrees /360 degrees) × π × (24 units)².
- Simplify the fraction and perform the multiplication to get the final area.