From a point P, on the surface of radius 3 cm, two cockroaches

Question

From a point P, on the surface of radius 3 cm, two cockroaches A and B started moving along two different circular paths, each having the maximum possible radius, on the surface of the sphere, that lie in the two different planes which are inclined at an angle of 45 degree to each other. If A and B takes 18 sec and 6 sec respectively, to complete one revolution along their respective circular paths, then after how much time will they meet again, after they start from P?

A.27 sec
B.24 sec
C.18 sec
D.9 sec
E.7 sec

A.27 sec
B.24 sec
C.18 sec
D.9 sec
E.7 sec

chetan2u · Answer

It doesn't matter what angle they are to each other or what is the radius of the sphere, their path will intersect at only two opposite points, one at P and other exactly opposite..
For example, say P was North pole, so the other intersecting point will be South pole...

Now, time to reach the opposite point will be SAME as covering half revolution, so 18/2 or 9 and 6/2 or 3..
So they will meet first time after TIME that will be the LCM of (9,3) or 9

D

garcmillan · Answer

Hi all,

This is a very good question that shows how GMAT can provide data that may not be necessary to use to solve the question:

the 45 degree inclination is needed to understand that they can only meet at P and its opposite point (180 degrees), but it does not matter the specific inclination (it could be 30 degree inclination and the answer would be the same). So this info just tells us that they can meet at point P or at its opposite (i.e. half of a revolution for A and B)

Intersection of point A with P/opposite of P:   9 , 18, 27, 36... (starting at 9 seconds and every 9 seconds)
 Intersection of point B with P/opposite of P:  3, 6,  9 , 36... (starting at 3 seconds and every 3 seconds)

They would intersect after 9 seconds in the opposite of point P

ANSWER D

aaa707 · Answer

Got trapped choosing C : 18... was using LCM but started at

18, 36 ...

6, 12, 18, 24, 36 ...

can someone explain the flaw of my process?

JuliaKrupnik · Answer

Can you please explain, why the path will intersect at only two OPPOSITE points? I have difficulty visualizing these paths. 
Thank you in advance!!

PRhea · Answer

I am not able to visualise this. Could you please explain how the path will intersect at two points

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ashmitakhan · Answer

The information about the angle provided is unnecessary. Now, let's tackle the question. This is what I did!
I found out the relative speeds of the two ants. 
It says that they have different circular paths, meaning they will start with the opposite ends of the circle. But the total distance travelled 
will be the circumference. 
Hence, D= 2*22/7*3= 132/7
Now, Speed of A= 132/7/18= 22/21 cm/sec
Speed of B= 132/7/6= 22/7 cm/sec
Now, find their relative speed=
22/7- 22/21= 44/21

T= Speed/Distance= 132/7*21/44= 9 secs

Correct answer is D.

josenetofaria · Answer

They are running in opposite directions.

Then, their paths intersect in two points: the starting point and the diametrically opposite point.

So, it tuns out that the A runs 1 circle in 18 sec and the B runs 1 circle in 6 sec.

Visualize this: every time they meet, the sum of distance traveled is a multiple of 1 circle (notice that their paths have the same radius).

Mathematically, this means that the sum of distances should be an integer every time they meet.

So, D_A = t*(1 circle/18) and D_B = t*(1 circle/6).

Sum = D_A + D_B = 4t/18 = t/9. To be integer, the minimum t is 9.

Therefore 9 seconds is our answer.

Fdambro294 · Answer

I drew a picture with the 2 starting Points and Paths "inclined at 45 degrees."

Then it took me a bit to realize that if the 2 cockroaches meet, they will meet on the Complete OPPOSITE Side of Point P, which is exactly 1/2 * the Entire Surface Area of the Sphere. This will happen when they both meet at the SAME INSTANCE/TIME exactly 1/2 Revolution around the Sphere from their Starting Point.

This is True because both cockroaches are moving along the "maximum possible radius on the sphere" in opposite directions (which is just like making 1 Complete Revolution around the Middle of the Sphere from Starting Point P)

If it takes Cockroach A 18 seconds to complete 1 Full Revolution, then it will take Cockroach A 9 seconds to complete 1/2 Revolution around the Sphere to the Opposite Side of Starting Point P

In those 9 seconds, Cockroach B will complete 1 Full Revolution around the Sphere (it takes B 6 seconds to do so) and then another 3 Seconds to travel 1/2 a Revolution and reach the Opposite Side of Starting Point

Thus, in 9 seconds both Cockroaches will meet on the Opposite Side of the Starting Point P on the Sphere.

D

Seemed to easy to be True at first. Hard part is picturing the Sphere in your head and realizing that if they meet, it can only be at the Point that is exactly 1/2 Revolution around the Sphere from Starting Point P OR Starting Point P again.

I pictured a Small Miniature Earth and 2 Airplanes flying around the Entire Earth Sphere.

dungeon123 · Answer

I did understand the reasoning provided, but I have trouble grasping the question! What does the question mean by ‘moving along two different circular paths, each having the maximum possible radius, on the surface of the sphere, that lie in the two different planes which are inclined at an angle of 45 degree to each other.’ Can you explain it through diagram?

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ShreyasJavahar · Answer

The maximum possible radius simply means A and B skim along the entire circumference of the sphere along two different paths. Picture the Earth for example, let's say A goes from north to south pole and B goes along the equator. You should be able to see that these paths trace a circle of "maximum radius", maximum radius here just means radius. Just think of it as two circles of equal radius joined at two points, the sphere bit isn't really relevant, the inclination of the two cross-sections(the paths that A and B take) is also irrelevant. You can see in the diagram below that the angle x could be anything, the distance that would have to be covered by A and B would be the same regardless of the angle.
The two tiny circles mark the only two points of intersection of the paths that A and B take.

StacyArko · Answer

Thanks very much for this picture. Helps a lot with visualization. I see 4 points of intersection though. Why are we not considering the other two?

And if we include the other 2 points of intersection and mark the time A&B reach each of the 4 points, will the time they meet be 4.5 instead of 9 (I acknowledge 4.5 is not an answer option, so this is just to understand the concept). Thanks in advance.

ShreyasJavahar · Answer

There are only two points of intersection, you'll have to picture the diagram in a 3 dimensional plane. The other two points you are referring to are not points of intersection. Simply imagine two rings on a globe, one running from the north pole to the south pole, and back to the north pole, and another running along the equator. These two rings intersect only at two places, not four. You could also try this with any two rings or bangles with similar radii, try to pass them through each other at any angle where they are not parallel, and you'll see only two points of intersection

Regor60 · Answer

The key to understanding why there are two points of intersection is in understanding what it means having two paths inclined.

First, we know the paths circle the center of the sphere because each is stipulated to be at the maximum radius.

Next we should understand that one path inclined to another means rotating one path relative to the other.

This rotation occurs around a line through the center.

Well, a line through the center of a sphere has two fixed points on the surface of the sphere that both paths intersect, like an axle.

From a point P, on the surface of radius 3 cm, two cockroaches

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From a point P, on the surface of radius 3 cm, two cockroaches

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