Final answer:
To find the central angle of a sector with an area of 6in² and a radius of 5in, the formula for the area of a sector is used. The measure of the central angle is calculated to be approximately 27.5 degrees.
Step-by-step explanation:
The student is asking to determine the measure of the central angle of a sector with a given area and radius. To find this angle measure in a circle, we use the formula for the area of a sector, which is Area of Sector = (θ/360) × π × r², where θ is the central angle in degrees, r is the radius, and π is the mathematical constant Pi (~3.14159).
Given:
Area of sector = 6 in²
Radius (r) = 5 in
We can solve for θ:
6 = (θ/360) × π × 5²
6 = (θ/360) × π × 25
6/(π × 25) = θ/360
(6/π × 25) × 360 = θ
θ = (6 × 360)/(π × 25)
θ ≈ (2160)/(78.54)
θ ≈ 27.5°
The measure of the central angle is approximately 27.5 degrees.