Final answer:
To find the length of XY, we can use the formula for the area of a sector of a circle. By substituting the values, we can solve for the radius and then find the length of XY.
Step-by-step explanation:
To find the length of XY, we need to use the formula for the area of a sector of a circle. The formula is A = (1/2)r^2θ, where A is the area, r is the radius, and θ is the central angle in radians.
Since the area of the shaded sector is given as (4/3)π, we can equate this to the formula and solve for θ. (4/3)π = (1/2)r^2θ.
Since ∠XWY = 30° = π/6 radians, we can substitute the values and solve for r. (4/3)π = (1/2)r^2(π/6).
Simplifying, we have 4π/3 = πr^2/12. Cross multiplying, we get 48π = 3πr^2. Dividing both sides by 3π, we get r^2 = 48/3 = 16.
Taking the square root of both sides, we get r = 4. The length of XY is equal to the diameter of the circle, which is 2r. Therefore, XY = 2 * 4 = 8. Hence, the length of XY is 8π.