Jump to content
BrainDen.com - Brain Teasers
  • 0

Concentric circles on steroids



A goat is tethered to a circular silo of radius r = 29 yd that is centered in a circular grassy field of radius R. The goat is just able to keep the silo wall clear of grass. He is also able to eat half of the grass in the annular field.

How big must that field be?

Link to comment
Share on other sites

10 answers to this question

Recommended Posts

  • 0

The goat in the picture is about 50 yards long and 40 yards tall. I would not like to meet it in a dark alley.

For any rope length, a field could be built, such that the goat would have access to half of it. I guess, we are looking for the minimum size field.

The minimum rope should allow the goat to reach around half of the silo. Thus the rope length is 29π yards.
After that, it seems, the problem requires solving an integral.

Edited by Prime
Link to comment
Share on other sites

  • 0


Although it's entirely possible I made a math error somewhere *shrugs*.

I'm too sick/lazy to make a pretty diagram, but I divided the area the goat spans in half along the line of symmetry. The 'funky' part (i.e. where the leash is at least partially along the silo wall) can be described by the function r(theta)=29(pi-2*theta+2sin(theta)). To get the area, integrate 1/2*r(theta)^2 over theta from 0 to pi/2. Double that and subtract the area of the silo. The part where the goat's leash does not touch the silo is a semi-circle of radius 29.

The rest is a little geometry, a little algebra.
Link to comment
Share on other sites

  • 0

I am wondering whether my analysis is off a bit.

I get a slightly smaller R than Y-San does.
But RG's answer gives exactly twice hers, meaning he may have given diameter instead of radius.
So there is the possibility they are both correct (with a misplaced factor 2 in RG's case - a typo more or less.)

Anyone else want to give it a shot, I'll go back over my equations meantime.

Nice going, both, in any event. ;)

Prime is correct, it involves an a messy integral.

And to be clear, the area the goat covers is half the annular field of grass.

Link to comment
Share on other sites

  • 0

If i got the question right, then

R is approx. 125.54

dm92 your answer answer agrees with that of several others.

Y-san and I get a slightly smaller number.

It puzzles me (npi) what is the difference in solving this that leads to two different solutions.

Maybe the wording of the OP is ambiguous.

R = 121.14 yd
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.


  • Recently Browsing   0 members

    • No registered users viewing this page.
  • Create New...