So, if you think about it, that area is the difference between the areas of the two circles. It's the area covered by the larger circle that's not coverd by the smaller circle.
***The area of the larger circle - the area of the smaller circle = the area of the shaded region.***
Now to find the areas of the circles:
Well, we know that the formula for the area of a circle is: A=pi times the radius squared. (A=πr^2). You know that the radius of the bigger circle is R. Since the radius of the smaller circle is 3 units less, the radius of that circle is R-3 (makes sense?).
So know you plug those values into the area formula:
The area of the bigger circle is A=πR^2 (since r, the radius, in the formula is R in the bigger circle)
In the same way, the area of the smaller circle is A=π(R-3)^2.
Now, plugging those into the starred formula,
*********The area of the shaded region is πR^2 - π(R-3)^2.************
You can factor this like so: π[R^2 - (R-3)^2]. (All I basically did was pull π out of both terms)
Now to simplify the expression inside: (R-3)^2 = (R-3)(R-3) = R^2 -6R + 9 (use FOIL, or if you have the squaring formula memorized, that helps).
So now, the expression for the area of the shaded region should look like this:
π[R^2 - (R-3)^2] becomes π[R^2 - (R^2-6R+9)] = π[R^2 - R^2+6R-9] (distribute the negative inside the parentheses)
Then, subtracting inside the parentheses: π[R^2 - R^2+6R-9] = π[6R-9]
Finally, factoring a 3 out of the inside parentheses, you get:
The area of the shaded region = 3π(2R-3)
Hope that helps! Let me know if you need more help!!!
(Response) Hey, sorry I didn't get back to you earlier.
I don't really think that C can be the answer unless the radius of the smaller circle is 3. If that's so, then you can plug 3 into the formulas for the smaller circle instead of R-3.
But since you said, "the radius of the smaller circle is 3 units less than R", the best answer up there seems to be D (the place where it is in my explanation is now starred, along with the "formula" I used to get it.)
I don't mean to act stuck-up by saying this, but I think the book may be wrong if it's saying C. The only reason I'm saying that is because a c (her answer is above mine, I think) used the same idea as I did, only I think she forgot to distribute the negative when she subtracted at the end.
Best of luck!!! If you need any more help, just mention it in the question... and I'll be sure to check this time! :P