I encountered this problem on one of the tests. It was around my 12th or 13 question. I do admit that this question stumped me and I took some amount of time in solving this question.
I started off with the geometric approach but then quickly got tangled in it. I then shifted to approximating the area and using the process of elimination. I am explaining my approach even though it is unconventional and based only on the situation that I was in.
assumption pi=3.14
The radius is given as (OR/2)=9. Hence the circumference of the circle is 18pi.
Clearly looking at the figure, the length of arc PQ is much smaller than 1/4 the circumference.
18*pi=18*3.14~42+ (read it as something greater than 42)
Hence 1/4 of the circumference is ~ 10+ (read as something greater than 10). The length of Arc PQ is much smaller than 1/4 of the circumference as well.
Options C and D can be eliminated since C=~10.5 and D=~13.5. These can be clearly eliminated. Similarly Option E : 3pi ~ very close to 10 and hence I eliminated it.
After this it was purely on the figure given than I eliminated the other choices. The arc PQ looks to be more like closer to half of the 1/4 i.e. ~5+. I really have no justification as to how I deduced it. Just an observation so that I could guess the answer.
Option B [(9pi)/4] comes to be roughly 7- (read as less than 7)
Option A (2pi) comes to ~6.5- (read as less than 6.5) which is closer to mu initial estimate of being 5+.
Hence I chose A.
I am very well aware that on some other day it is very much possible to choose B and to get the question wrong. Also I am not advocating this method in any way
Its just that this was an educated guess which I guessed had 50% chance of being right.
The above explanations and diagramatic representations have helped me realize my mistake and should such a problem appear again I am dead sure of the mathematical way to solve it.
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My attempt to capture my B-School Journey in a Blog : tranquilnomadgmat.blogspot.com
There are no shortcuts to any place worth going.