In the figure, the area of the shaded region is 224π cm^2.

what is the value of x?

(my answer is 7)

Here is my take... I'm getting x = 10 as an answer.  Here was my approach.  

Overall: the area of the large circle minus the area of the small circle will give the area of the shaded region.. the following is how this looks mathematically:

Recall: Area of a circle is defined by πr² and I will be using 'x' as the radius of the smaller circle.  So:
324π is the area of the large circle and (x)²(π) represents the area of the smaller circle:  

Now 324π - x²(π) = 224π
or,  324 - x² = 224
or x² = 100
and now you can see x = 10

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