wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> hard >> Fence MAX
        
(Message started by: Wonderer on Apr 23^rd, 2007, 9:30pm)

Title: Fence MAX
Post by Wonderer on Apr 23^rd, 2007, 9:30pm

There’s a circular farm. The area of the farm is 2000/3 m^2. You are given 10 pieces of wooden boards to form a fence inside the farm. Here are few rules:
1 each wooden board is 1.5m in length and 0.2m in width
2 the maximum distance between two wooden boards is 1 meter
3 the fence and part of the fence’s perimeter must form a closed loop

Question
What method should you use to maximize the area within the fence and farm perimeter? And what is the area?
(Are you sure you used the wooden boards in the most effective way?)

Title: Re: Fence MAX
Post by jollytall on Apr 24^th, 2007, 12:02am

Help me understanding the question, please:

- Is the farm already fenced?
- Do we have to fence around an inside area of the farm using the existing fence AND the 10 boards? This assumes the farm has already got a fence.
- If not, then I do not see the relevance of the size of the farm.
- Do I understand correctly, that the fence built from the boards is considered a closed fence if the maximum distance (gap) in between two is 1 meter? It would mean that you make a fence against e.g. cars but not people or animals.
- Or, do you mean that any two boards of the 10 must be closer (or equal to) than 1 meter?

Title: Re: Fence MAX
Post by Grimbal on Apr 24^th, 2007, 12:49am

I sense a solution where all the questions but #1 and #2 don't really matter. ::)

on 04/23/07 at 21:30:34, Wonderer wrote:

Are you sure you used the wooden boards in the most effective way?

Yes.

Title: Re: Fence MAX
Post by Wonderer on Apr 24^th, 2007, 1:33am

- Is the farm already fenced? YES
- Do we have to fence around an inside area of the farm using the existing fence AND the 10 boards? This assumes the farm has already got a fence. YES
- Do I understand correctly, that the fence built from the boards is considered a closed fence if the maximum distance (gap) in between two is 1 meter? It would mean that you make a fence against e.g. cars but not people or animals.
YES
Excuse my poor English. :0

Title: Re: Fence MAX
Post by jollytall on Apr 24^th, 2007, 3:06am

Well, I would go for the whole area of the farm. I just build the extra fence along the old fence, or somewhere inside the farm would make a zero area fenced part. In both cases the area between the perimeter and the new fence is equal to the original area of the farm.

Probably you should ask that when the inner fence is built and splits the farm into two parts, what is the maximum, the smaller area can be?

Title: Re: Fence MAX
Post by Wonderer on Apr 24^th, 2007, 6:22am

when the inner fence is built and splits the farm into two parts, what is the maximum, the smaller area can be?

Title: Re: Fence MAX
Post by Grimbal on Apr 24^th, 2007, 8:51am

I find a diameter of 50.46 meters.

The optimal way of fencing would be to use the max length of sqrt(1.5^2+0.2^2) meters diagonally for each plank. That means to place the planks almost horizontally but not quite. To this length you would add 11 gaps of 1 meter. I assume that the 1m gap rule also holds between the planks and the outer fence. That makes a total length of 26.1327 altogether, which is unfortunately not enough to split the farm in 2 equal parts.
So, I would arrange these around a circular path perpendicular to the outer fence. I would bend the planks to better follow the optimal shape of the circle. The interesting thing is that if I arrange the available length around a circle, I get curved 1m gaps, which would mean that the planks are in effect less than 1m apart. So, I should measure the gaps along the cord. But then I have the situation that theoretically the fenced area (the circular path) exceed a bit the area defined by the straight lines connecting the planks. Is that correct? If yes I would abuse it and add for example 1/2 disk more in front of each gap as "fenced in". I would just draw a very complicated fence line that include more area without moving the planks (and keeping the distance between them). I could cover the whole surface. If I have to follow straight lines, a circular path is not optimal any more. The planks should be bent a bit more, but I don't know exactly by how much.

It looks complicated.