A castle is at the center of the several flat paths which su

Question

CastlePaths1.gif 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

kundankshrivastava · Accepted Answer

Let's take distance from castle to circular moat is r.
side of square = sqrt2 * r
Perimeter of Circle = 2* Pi * r

Attachment:

File comment: Castle Paths

gmatcastle.jpg [ 32.43 KiB | Viewed 13253 times ]

Total length of paths=q = 4* r + 4 *sqrt2 *r +2*pi*r

q = (4+4*sqrt2 + 2* pi) r
or , r = q/(4+4*sqrt2 + 2* pi)
or, r = q/2(2+2*sqrt2+Pi)

Hence, Answer is B

crejoc · Answer

i couldnt figure out how to find the perimeter of the square in terms of r...

bhushan252 · Answer

OA B

let distance from the castle to the circular moat is x

distance from 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 ( addition of lenght of diagonals of sqr)= 4x

Thus side of inscribed sqr = x*sqrt2 (use pythagoras 2x^ = (side of sqr)^2.....hence perimeter of sqr = 4*sqrt2*x

circumference of perimeter = pi * (2*x)
 
Thus our eqn becomes => q= pi * (2*x) + 4*sqrt2*x + 4x

on simplification we get x = q / 2(2+2*sqrt(2)+pi) km... OA B

crejoc · Answer

Thanks for the solution...

rathoreaditya81 · Answer

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. If the total length of all of the pathways is q kilometers, then which expression represents distance from the castle to the circular moat?

1) q/4(2+2\sqrt{2}+pie)
2) q/2(2+2\sqrt{2}+pie)
3) q/(2+2\sqrt{2}+pie)
4) 2q/(2+2\sqrt{2}+pie)
5) 4q/2(2+2\sqrt{2}+pie)

Attachments

CastlePaths1.gif [ 2.33 KiB | Viewed 16009 times ]

1) q/4(2+2\sqrt{2}+pie)
2) q/2(2+2\sqrt{2}+pie)
3) q/(2+2\sqrt{2}+pie)
4) 2q/(2+2\sqrt{2}+pie)
5) 4q/2(2+2\sqrt{2}+pie)

rathoreaditya81 · Answer

Strange, this problem seems to have a different answer when we take the side of the square to be a ( or r) or may be I am doing something wrong!
Here, what I am doing -

Let the side of the sq be a, then sum of four sides = 4a;
Using formula, diag of a sq is sqrt(2)a, we get sum of the paths four paths from castle to moat ( i.e, sum the diagonals) as = 2a sqrt(2) ;
Circumference of the circle as 2pie a sqrt(2)/2

Putting it all together we 
4a +2asqrt(2) +2pie sqrt(2)/2 = q
=> 4a +2asqrt(2) +pie asqrt(2) =q
=> sqrt(2)a (2 sqrt(2)+2 +pie) = q
=> a = q/sqrt(2) (2 sqrt(2)+2 +pie)

Can anybody shed some more light on this..?

swatirpr · Answer

Let take radius of circular path is R
so distance from the castle to the circular moat = R

total length of all of the pathways = total length of 4 straight paths that travel from the castle to its circular moat + length of circular path which borders the moat + total length of square path which has its corners at the ends of the 4 straight paths

(Diagonal of square is 2R, (90-45-45 right triangle with hypotenuse 2R) so side of square would be  R\sqrt{2})

q = 4R + 2R*pie + 4 (R \sqrt{2}) 
 q = 2R (2+pie+2 \sqrt{2}) 
 R = q/2 (2+2 \sqrt{2}+ pie)

Answer B

rathoreaditya81 · Answer

Hi Swatipr, have a look at - castle-paths-82093.html#p653668

rathoreaditya81 · Answer

Thanks a lot swatipr.. oblivious to obvious..strange manifestation of alcohol after effect! :twisted:

ruturaj · Answer

can we solve this in more simpler way..say plugging smart nos

TooLong150 · Answer

This problem is not that difficult once you understand the question; it just takes too long. Can anyone explain how to do this under 3 minutes?

KarishmaB · Answer

You can actually do it in a minute after you understand that all you need is the radius of the circle given that sum of lengths of all lines and curves here is q.

So q is made up of the circle, the inscribed square in it and the 4 radii of the circle. Assume radius of the circle is r.
Each side of the square will be  \sqrt{2}r  because each side makes an isosceles right triangle with two radii. Each radius is r and sides of isosceles right triangle are in the ratio  1:1:\sqrt{2}

q = Circumference of circle + Perimeter of square + 4*radius
 q = 2 \pi r + 4\sqrt{2}r + 4r

So  r = q/2( 2 + 2\sqrt{2} + \pi)

Answer (B)

russ9 · Answer

Hi Karishma,

The part that threw me off was the  \sqrt{2}r

I understood that the hypotenuse was  \sqrt{2}  but I couldn't connect the R there, and more importantly, I couldn't tell that the problem was asking us to solve for R.

Any thoughts on how I would go about this in the future?

mvictor · Answer

wow..the wording is very confusing...
total pathways : circumference, the square, and 4 radii.
circumference is 2*pi*r
square - since we have 4 equilateral triangles, and all right triangles, we can apply 45-45-90 property, and get one side of the square equal to 4*r*sqrt(2)
then we have 4r
we are then told that:
q=2*pi*r + 4*r*sqrt(2) + 4r
factor r
q=r(2pi+4*sqrt(2)+4)
r= q/(2pi+4*sqrt(2)+4)

we can eliminate D and E right away
from the above, we can factor out a 2;
q/2(pi+2*sqrt(2)+2)

so B.

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