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In the figure above, an unshaded circle with diameter x is surrounded by a shaded region with uniform width of 2. For which of the following values of x is the area of the unshaded circle greater than the area of the shaded region?
I. 9
II. 10
III. 11
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
Attachment:
1 (1).jpg [ 5.74 KiB | Viewed 1024 times ]
28 May 2022, 18:49
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In the figure above, an unshaded circle with diameter x is surrounded by a shaded region with uniform width of 2. For which of the following values of x is the area of the unshaded circle greater than the area of the shaded region?
I. 9
II. 10
III. 11
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
Attachment:
1 (1).jpg
Before we start doing any "real" work, let's think about it a little. If the unshaded region is tiny, the shaded will be larger than the unshaded. If the unshaded region is huge, the shaded will be smaller than the unshaded. There's some point in between tiny and huge where they will be equal.
Okay, let's just try 10 and see if that tells us anything. If the diameter of the unshaded area is 10, the radius is 5. That makes the radius of the larger circle 7. The area of the unshaded is 25pi. The are of the larger circle is 49pi. That makes the area of the shaded region 24pi. Is the unshaded greater than the shaded? Yep, by juuuuuuust a little. We need II in our answer choice. A is out.
We already know that as the unshaded area grows, it will be larger than the shaded region, so anything larger than 10 will also work. We need III in our answer choice. B and C are out.
10 was just BARELY large enough. 9 is definitely going to be too small. We can't have I in our answer choice. E is out.
Answer choice D.
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