If O and P are each circular regions, what is the area of the smaller

Question

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

SKMishra · Answer

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.

ashokk138 · Answer

Let the radius of Circle O be R and P be r

St 1:

π* R^2 - π*r^2 = 21π
R^2 - r^2 = 21
(R+r)(R-r) = 21

We do not know anything about R or r, hence Not Sufficient.

St 2: 
2πR - 2πr = 6π => R - r = 3. Not sufficient

Combining 1 and 2,
R+r = 7 and R-r = 3
Solving we can find R = 5 and r = 2. 
Hence we can area of the smaller region.
Sufficient.

Option C.

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susheelh · Answer

Hello!

I have a question on this problem. Can someone please help? I think the answer is E

The stem essentially wants us to tell Which circle O or P has smaller radius.

Let the Radius of O = R and Radius of P = r

S1 is essentially saying  R^2 - r^2 = 21 \ OR \ r^2 - R^2 = 21 . This is clearly insuff

S2 is essentially saying  R-r = 3 \ OR \ r - R = 3 . This also is clearly insuff

S1+S2:

Case1:  R^2 - r^2 = 21  and  R-r = 3 , we get R = 5 and r = 2. Answer = P
Case2:  r^2 - R^2 = 21  and  r - R = 3 , we get r = 5 and R = 2. Answer = O
Two possible answer and hence insuff.

Final answer should be E.

Bunuel  or other math experts, mind having a look and correcting me please!!

Thanks in advance.

Bunuel · Answer

What question are you answering? The question is: what is the area of the smaller of these regions?

srikanth9502 · Answer

Stmnt1) pi(R^2-r^2) = 21pi - Insuff

Stmnt 2) 2pi(R-r) = 6pi -> R-r = 3 - Insuff

S1 & S2 ) 
Pi(R+r)(R-r) = 21pi
(R+r)3 = 21
R+r=7 ----Eqn1
R-r = 3----Eqn 2(From S2)

Solviving 1 & 2, R=5
So, area is 25pi

susheelh · Answer

Thanks Bunuel for the reply! My Bad, the question is asking the smaller of the two Areas. Not which "is" the The Smaller circle.

I get it now. Lesson Re-learnt for me- Read carefull!. :cry:

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If O and P are each circular regions, what is the area of the smaller

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