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

[mitzwillrockgmat]

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?



Hello! can anyone shed light on this problem? thanks :0)

[liferocks]

Let distance from the castle to the circular moat be a
then radius of the circle is a
side of the square is aV2

perimeter of square=4aV2
perimeter of circle is = 2*pi*a

and total length of diagonals of square is 4a

hence the total length of all of the pathways is =4aV2+2*pi*a+4a=2a(2V2+2+pi)=q
so distance from the castle to the circular moat=a=q/{2(2V2+2+pi)}..option B

[mitzwillrockgmat]

Let distance from the castle to the circular moat be a
then radius of the circle is a
side of the square is aV2

perimeter of square=4aV2
perimeter of circle is = 2*pi*a

and total length of diagonals of square is 4a

hence the total length of all of the pathways is =4aV2+2*pi*a+4a=2a(2V2+2+pi)=q
so distance from the castle to the circular moat=a=q/{2(2V2+2+pi)}..option B

im a bit confused with this one. aren't the four path from the castle, half the diagonals of the square? diagonal of square is side X sq root 2. hence these four paths should be if i name each of the square a, each path is [(a X sq root 2) / 2] no ?

[indiantiger]

Let the side of the square be = a
then the diagonal is sqrt(2)*a
perimeter of circle = IId (where d = sqrt(2)*a)
total distance = q
q = perimeter of square + perimeter of circle + length of two diagonals
q= 4a+ II*sqrt(2)*a+ 2*(sqrt)*a
q = a(4+II*sqrt(2)
Let the side of the square be = a
then the diagonal is sqrt(2)*a
perimeter of circle = IId (where d = sqrt(2)*a)
total distance = q
q = perimeter of square + perimeter of circle + length of two diagonals
q= 4a+ II*sqrt(2)*a+ 2*(sqrt)*a
q = a(4+II*sqrt(2) + 2*sqrt(2))
a = q/{4+II*sqrt(2) + 2*sqrt(2)}

Now the expression that represents distance from the castle to the circular moat

= 1/2 * length of either diagonal
= 1/2 * sqrt(2)* a
= 1/2 *sqrt(2) * q/{4+II*sqrt(2) + 2*sqrt(2)}
take sqrt(2) common from the denominator
= 1/2 * sqrt(2) *[q/sqrt(2){2sqrt(2)+II+2}]
= q/2{2+2sqrt(2)+II} (ANS : B)