Answer: Radius = 6*sqrt(3) meters
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Work Shown:
The unshaded portion is 310 degrees, while the shaded portion is 360-310 = 50 degrees
This means 50/360 = 5/36 of the full circle area is the shaded region of 15pi square meters
So,
s = shaded area = 15pi
s = (5/36)*(full circle area)
s = (5/36)*(pi*r^2)
(5/36)*(pi*r^2) = 15pi
(5/36)*r^2 = 15
5r^2 = 15*36
5r^2 = 540
r^2 = 540/5
r^2 = 108
r = sqrt(108)
r = sqrt(36*3)
r = sqrt(36)*sqrt(3)
r = 6*sqrt(3)