noTh1ng wrote:
If O and P are each circular regions, what is the area of the smaller of these regions?
1) The difference between the areas of the regions O and P is 21π
2) The difference between the circumferences of regions O and P is 6π
Statement 1:
If the difference between the areas is \(21\pi\), then smaller area could be anything. For example, if smaller area is 2π, larger area is 23π; and if smaller area is 4π, larger area is 25π. Hence insufficient.
Statement 2:
Same reasoning as in the previous statement, smaller area could have any radius. Hence, insufficient.
Combined:
Statement 1 gives \(\pi{R}^{2} - \pi{r}^{2} = 21\pi\), which implies that \({R}^{2} - {r}^{2} = 21\), i.e. (R - r)(R + r) = 21.
Statement 2 gives \({2}\pi{R} - {2}\pi{r} = 6\pi\), which implies that R - r = 3
From the above two: R - r = 3 and R + r = 7, which gives r = 2.
Hence, the area can be determined using both statements together.