Solved

There are four dogs/ants/people at four corners of a square of unit distance. At the same instant all of them start running with unit speed towards the person on their clockwise direction and will alw

Posted on 2011-02-21
9
1,847 Views
Last Modified: 2012-05-11
There are four dogs/ants/people at four corners of a square of unit distance. At the same instant all of them start running with unit speed towards the person on their clockwise direction and will always run towards that target. How long does it take for them to meet and where?
0
Comment
Question by:dshrenik
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 6
  • 2
9 Comments
 
LVL 39

Expert Comment

by:Aaron Tomosky
ID: 34947937
They meet in the center in 1/2 unit time.
0
 

Author Comment

by:dshrenik
ID: 34947952
If possible, can you give an explanation.
I think they move in spirals.. but why does it take 1/2 unit time?
0
 
LVL 39

Expert Comment

by:Aaron Tomosky
ID: 34947986
It's actually slightlymore that 1/2 unit time because it's an arc. So it's really 1/4 of the perimeter of a circle with a radius of 1/2 unit. pie*1/2 unit squared /4
Sorry that the best ican type on my phone.

So let's look at bottom left and bottom right. Bl starts moving up and br starts moving left, but see how br starts curving up since Bl is going up? Bl is also curving right since tl is moving right. So they all move in a nice arc toward the center
0
Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

 
LVL 37

Expert Comment

by:TommySzalapski
ID: 34948009
If they travel in perfectly circular paths (which I am not yet sure about, but it seems probable) then they would travel along a circle centered at the midway point on the edge of the side of the square.
Then they would travel a distance of pi/16 units which is what aarontomsky was trying to say.
pi/16 since the whole circle is pi*r^2 r is 1/2 unit and it's 1/4 of the circle so pi(1/2)^2*1/4 = pi/16
0
 
LVL 37

Expert Comment

by:TommySzalapski
ID: 34948015
What? Oops. That's area. It's actually pi*d/4 so pi/4 since d is the unit side. My bad.
pi/4 is the distance travelled.
0
 
LVL 37

Expert Comment

by:TommySzalapski
ID: 34948031
Scratch all that. They would not move in perfect circular curves. Try it out. Draw the square and the circles and you can see that at the halfway point the paths would not point at each other. I'll have to think about it for a while.
0
 
LVL 37

Accepted Solution

by:
TommySzalapski earned 500 total points
ID: 34948322
The total distance travelled is 1. 1 unit is travelled.

Think about it like this. If you go 1/2 of the way before noticing that the other guy moved then you will travel 1/2 unit. Now look. They are all still in a square (but it's a rotated square) and the side of the new square is sqrt(2)/2 (pythagorean theorem) since the diagonal of the square is 1/2.
 step1So the solution for square of size 1 is S(1)
S(1) = 1/2 + S(sqrt(2)/2)
This is 1/2(1 + r^2 + r^3 + r^4 ...) where r is sqrt(2)/2.
Now if r > 0 and r < 1 then 1 + r^2 + r^3 + ... is a geometric series and equals 1/(1-r)
So in this case the answer is 1 + sqrt(2)/2
But what if we go only 1/3 of the way?
Using the pythagorean theorem again it is 1/3(1 + r^2 + r^3 ...) where r is now sqrt(5)/9
So the geometric series gives 1/3(1/(1-sqrt(5)/9)) = 3/(9 - sqrt(5))
So what if we go 1/n of the way and n tends toward infinity?
1/n(1 + r^2 + r^3 + r^4....) where r is sqrt(n^2 - 2n + 2)/n
Again geometric series makes it 1/n(1/(1-r)) or 1/(n - sqrt(n^2 - 2n + 2))
And (if you know limits from calculus) as n goes to infinity that goes to 1
0
 
LVL 37

Expert Comment

by:TommySzalapski
ID: 34948325
If you don't believe it. Try plugging it into a calculator. Use bigger and bigger numbers for n.
0
 
LVL 37

Assisted Solution

by:TommySzalapski
TommySzalapski earned 500 total points
ID: 34948359
If you didn't catch what happened as n tended to infinity this is what I was doing.
If you go halfway to the other guy each time you got approximately 1.7071 as the answer.
If you went 1/3 of the way before turning you got ~ 1.309 as the answer.
What if you keep turning constantly? Basically (applied math guys cover your ears) you go 1/infinity of the way before you turn. So you are always turning. So basic calculus limits get us the answer.
Here is an Excel spreadsheet for those who don't know limits. The number in column B is the portion of the distance you go before turning. Note how if you go the whole way you get an error. This is because you would end up at the other corner and would never get closer to the middle. So the math even works at the extreme ends.
SpiralIn.xls
0

Featured Post

Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Suggested Solutions

Title # Comments Views Activity
using differentials to calculate area percentage error 8 726
Translate some English into Latin 5 362
xyzThere java challenge 153 192
Zyrtec oral for under 5 years old kid 6 90
Adults who share images on social media aren’t the only ones who need to worry about their privacy. Our culture’s tendency to share every move and celebration affects the privacy of our children, too.
In order to fulfill our mission of inspiring learning in the technology community, Experts Exchange is launching a Course of the Month program. Premium and Team Account members will have access to one course per month as a part of their membership, …
Email security requires an ever evolving service that stays up to date with counter-evolving threats. The Email Laundry perform Research and Development to ensure their email security service evolves faster than cyber criminals. We apply our Threat…
Attackers love to prey on accounts that have privileges. Reducing privileged accounts and protecting privileged accounts therefore is paramount. Users, groups, and service accounts need to be protected to help protect the entire Active Directory …

749 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question