I'm going to go ahead and answer in case someone has the same question.
The value of x would be 100°.
Since the radius of the large circle is 10, the area would be π(10²)=100π cm.
Since the radius of the small circle is 8, the area would be π(8²)=64π cm.
The area of just the ring would be the difference between these two:
100π-64π = 36π cm.
Since we are given that the area of the shaded region of the ring is 10π cm, we can use this to find what portion of the total it is.
10π/36π = 5/18.
The shaded part of the ring is 5/18 of the total. This means that the measure of the angle is 5/18 of the total:
5/18(360) = 100°
