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manhattan cat geometry question help me.

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by aykutkaraalioglu » Fri Jun 22, 2012 10:17 pm
A castle is at the center of the several flat paths which surround it: 4 straight paths that travel from the castle to its circular moat, where they meet up with a perfectly circular path which borders the moat; that circular path circumscribes a square path which has its corners at the ends of the 4 straight paths-see the diagram to the right. If the total length of all of the pathways is q kilometers, then which expression represents distance from the castle to the circular moat?
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by gmat_and_me » Fri Jun 22, 2012 10:25 pm
If the distance from castle to the moat is r, then the
total length of all the pathways =
(length of the circular pathway) + (length of the square pathway) +
(length of the four straight pathways from the castle to the moat) =

(2*pi*r) + 4*(root(2)*r) + 4*r = q

Solve for 'r' from this equation and you should get the
answer.

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by Anurag@Gurome » Fri Jun 22, 2012 10:30 pm
All distances and lengths are in kilometers.

Say, radius of the circle is r.

Length of each path of the 4 straight paths = r
Total length of the 4 straight paths = 4*r

Length of each side of the square = (√2)*r
Hence, total length of the square path = 4*(√2)*r

Total length of the circular path = 2Ï€r

Hence, total length of all the pathways = (4r + 4√2r + 2πr) = (4 + 4√2r + 2π)r

So, q = (4 + 4√2r + 2π)r
--> r = q/(4 + 4√2r + 2π)

Hence, distance from the castle to the circular moat = r = q/(4 + 4√2r + 2π)
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by GMATGuruNY » Sat Jun 23, 2012 4:06 am
aykutkaraalioglu wrote:A castle is at the center of the several flat paths which surround it: 4 straight paths that travel from the castle to its circular moat, where they meet up with a perfectly circular path which borders the moat; that circular path circumscribes a square path which has its corners at the ends of the 4 straight paths-see the diagram to the right. If the total length of all of the pathways is q kilometers, then which expression represents distance from the castle to the circular moat?
Image

A) q / 4(2+2√2)+π) km
B) q / 2(2+2√2)+π) km
C) q / (2+2√2+π) km
D) 2q / (2+2√2+π) km
E) 4q / (2+2√2+π) km
When you post a question, please include the answer choices.
We can plug in values as follows:

Image

q = the sum of all of the lengths = 4(10) + 4(10√2) + 20π = 40 + 40√2 + 20π.
The distance from the center to the circular moat = 10. This is our target.
Now we plug into the answer choices q = 40 + 40√2 + 20π to see which yields our target of 10.

Only answer choice B works:
q / 2(2+2√2)+π)
= (40+40√2+20π) / 2(2+2√2+π)
= 20(2+2√2+π) / 2(2+2√2+π)
= 10.

The correct answer is B.
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by gmatdriller » Fri Dec 26, 2014 11:10 pm
Actually this question is not a complex one; however, the inclusion of all path
lengths to mean circular path, flat(square) path, and straight path makes the idea convoluted.

Other than that, the question is simple.
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by [email protected] » Sat Dec 27, 2014 9:34 am
Hi gmatdriller,

You've come across one of the interesting aspects of word problems in the Quant section - they're often wordy because they have to provide enough explanation and "math legalese" so a Test Taker wouldn't be able to argue that the question was vague or "unfair" because of interpretive bias. Getting all of the necessary information on the pad and having a clear understanding of the actual question usually means that all that's left to do is a handful of math "steps" to answer the question.

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by gmatdriller » Sun Dec 28, 2014 7:58 am
Agreed Rich.
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