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Castle Paths

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Castle Paths

by vineetbatra » Mon Jul 13, 2009 12:23 pm
Please see the attached Question I got on MGMAT Exam.

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?

A) q / 4(2+2*sqrt(2)+pi) km
B) q / 2(2+2*sqrt(2)+pi) km
C) q / (2+2*sqrt(2)+pi) km
D) 2q / (2+2*sqrt(2)+pi) km
E) 4q / (2+2*sqrt(2)+pi) km
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Re: Castle Paths

by Stuart@KaplanGMAT » Mon Jul 13, 2009 1:12 pm
vineetbatra wrote:Please see the attached Question I got on MGMAT Exam.

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?

A) q / 4(2+2*sqrt(2)+pi) km
B) q / 2(2+2*sqrt(2)+pi) km
C) q / (2+2*sqrt(2)+pi) km
D) 2q / (2+2*sqrt(2)+pi) km
E) 4q / (2+2*sqrt(2)+pi) km
First, let's simplify the question.

The distance from the castle to the moat is the radius of the circle. So, the question really is, "what's the length of the radius".

We know that q has 3 components:

1) circumference of circle;
2) perimiter of square; and
3) 4 radii of the circle.

We can express each of these pieces in terms of r.

1) circumference = 2(pi)r.
2) each side of the square is the hypotenuse of an isosceles right triangle with legs of r. So, each side of the square is r(root2). Therefore, the perimiter of the square = 4r(root2).
3) simply = 4r.

We know the sum of these is q, so:

q = 2(pi)r + 4r(root2) + 4r

We can factor r out of each piece on the right side to get:

q = r(2pi + 4root2 + 4)

and then divide both sides by the bracketed term to get:

q/(2pi + 4root2 + 4) = r

Sadly, this doesn't appear among the choices; happily, we can just factor a 2 out of the bottom to get:

q/(2(pi + 2root2 + 2)) = r

which matches choice B.
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by vineetbatra » Tue Jul 14, 2009 9:17 am
Thanks Stuart, I was not able t9o find the value of the sides of the Square.

Vineet
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by bhumika.k.shah » Sat Apr 03, 2010 11:57 am
Thanks stuart...
I was soo taken off by the complexity of the question during the practice test.....
Though now, after reading ur explanation, it seems soo simple. :-)
Stuart Kovinsky wrote:
vineetbatra wrote:Please see the attached Question I got on MGMAT Exam.

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?

A) q / 4(2+2*sqrt(2)+pi) km
B) q / 2(2+2*sqrt(2)+pi) km
C) q / (2+2*sqrt(2)+pi) km
D) 2q / (2+2*sqrt(2)+pi) km
E) 4q / (2+2*sqrt(2)+pi) km
First, let's simplify the question.

The distance from the castle to the moat is the radius of the circle. So, the question really is, "what's the length of the radius".

We know that q has 3 components:

1) circumference of circle;
2) perimiter of square; and
3) 4 radii of the circle.

We can express each of these pieces in terms of r.

1) circumference = 2(pi)r.
2) each side of the square is the hypotenuse of an isosceles right triangle with legs of r. So, each side of the square is r(root2). Therefore, the perimiter of the square = 4r(root2).
3) simply = 4r.

We know the sum of these is q, so:

q = 2(pi)r + 4r(root2) + 4r

We can factor r out of each piece on the right side to get:

q = r(2pi + 4root2 + 4)

and then divide both sides by the bracketed term to get:

q/(2pi + 4root2 + 4) = r

Sadly, this doesn't appear among the choices; happily, we can just factor a 2 out of the bottom to get:

q/(2(pi + 2root2 + 2)) = r

which matches choice B.
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