MGMAT #3: Castle Paths

[kartikdave]

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


This is how proceeded to answer the question.
Considering O as the center and the vertices of square as A, B, C & D
All the sides of the squares will be-
BC = AB = AD = DC = r * sqrt(2) where r = radius
ie OA = OB = OC = OD = r

1) And finally how to proceed from here to find the answer?

Now q = total length of all the path ways. Does this mean that-
q = all the 4 sides of square + the 2 diagonals of the square?

2) In the OE, q is termed as the perimeter of the circle + perimeter of square + diagonals of square. PLEASE EXPLAIN.

3) I could not completely understand the OE because the image of the final expression did not load properly.

Bunch of thanks

[africanchallenge]

if I can understand that q = the 4 path from the castle (that is the 2 diagonal of the triangle) the "circular path" in the stimulus represents the circuference of the circle, and finaly they talk about the "square path", that is the square itsef.

so let r be the radius of the circle, we are asked to find r.
circular path : 2*Pi*r
"diagonals" 4*r
"square path" 4*(r*sqrt(2))

so q = 4*r + 2*pi*r + 4*r*sqrt(2)

answer B is the one for me.

Hope this helps

[vineetbatra]

I do not understand how you found the value of the side of the Square. i.e. r*sqrt(2)).

Radius is r, so diagonal is 2r

Applying Pythagorean theorem

a^2 + a^ 2 = (2r)^2

= a^2(1+1) = 2r^2
= a^2 = ((2r)^2)/2
= a = 2r/underoot 2.

Can you please explain where I am going wrong.

Thanks

[esledge]

I must be honest--this problem bugs me. There are so many concepts overlapping: squares, diagonals, circles, Variables in the answer Choices. It seems a bit much, but anyway...

The most inconvenient aspect of this question is the variables in the choices, and the way that the given variable q is the sum of so many different lengths. So, the first thing I do is define q on my terms.

I draw the picture and label the lengths with smart, easy numbers.

I make the four radii equal 1. That makes the sides of the square path each sqrt(2). The circular moat path is thus 2pi*r = 2pi*1 = 2pi.

So my q is thus 4 + 4sqrt(2) + 2pi. The question asks for the distance from the castle to one of the corners, which I have defined as 1. Thus, the correct answer should equal 1 when we substitute q = 4 + 4sqrt(2) + 2pi.

At that point, the form of the answers is a big clue--they all have constant + constant*sqrt(2) + constant*pi in the denominator. It is fairly straightforward to see that the q in the numerator in B will cancel with the denominator in B, giving the target value of 1.

With this approach, the problem bugs me a little bit less. At least, the answer choices seem a little more fair to me. The shared expression in each of the denominators means that the only question is where a factor of 2 (or 4) should go.

I hope it helps you all, too.

[kelty.niles]

I think I'm having the same issue; even when I pick numbers, I'm having trouble getting the correct values for the square, maybe because I have my formula wrong.

Diagonal of a square = Side * Root 2
2 * radius (in the problem) = Side * Root 2
(2 * radius) / Root 2 = Side

How are we getting side = radius * root 2 ??

[tim]

Kelty, your final equation was

(2 * radius) / Root 2 = Side

Just simplify now and 2/root2 gives you root2

You can verify that 2/root2 = root2 by multiplying both sides by root2: 2 = root2*root2

[kalyan_tcs08]

I do not understand how you found the value of the side of the Square. i.e. r*sqrt(2)).

Radius is r, so diagonal is 2r

Applying Pythagorean theorem

a^2 + a^ 2 = (2r)^2

= a^2(1+1) = 2r^2
= a^2 = ((2r)^2)/2
= a = 2r/underoot 2.

Can you please explain where I am going wrong.

Thanks

vineet bhai...

in a triangle if both sides are 'r' then hypotensue is sqrt(2) not 2r.
thats where u went wrong

[mschwrtz]

ty kalyan_tcs08. good attention to detail there

[ritikakulkarni]

Hi,

I am close to the solution. One thing I do not understand is why add the circumference (2pir) to the whole calculation. Can someone please help me with that?

Thanks in advance.
Ritika

[tim]

What do you mean by "whole calculation"? What have you tried on this one, and where did you get stuck?

[rustom.hakimiyan]

I can get to the equation "q = 4*r + 2*pi*r + 4*r*sqrt(2)" but what is the trigger that tells to separate out the r?


I guess a sign could be that the answer choices are in terms of "q" but is there any wording in the question stem that states as such?

Thanks.

[RonPurewal]

The problem asks for the "distance from the castle to the circular moat". That's r.

[AlexanderP229]

Could anyone please help me understand what I am doing wrong here. Below I have included the three distances we need to add together to get Q along with their equations.

the circumference of the circle:

2piR

the two diameters:

4r

the perimeter of the square:

r^2+r^2= S^2

2r^2=S^2

sqrt(2r^2)=one side of the square

multiply by 4 to get the perimeter

4sqrt(2r^2)=Perimeter of the square

----


My problem is that in the solution, the perimeter of the square is given as 4rsqrt(2). Am I missing some key exponent or square root knowledge?

Thanks.

[RonPurewal]

√(2r^2)
= √2 • √(r^2)
= √2 • r
= r√2

each of these steps should be pretty much automatic, by the way. if there's any hesitation, you should do an internet search for something like "simplifying roots practice problems" and drill until it's instinctive.

[AlexanderP229]

The fundamentals... Thanks for the quick response Ron.