Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Goat

A small goat is tethered by a rope to a point at ground level on a side of a square barn which stands in a large horizontal field of grass. The sides of the barn are of length 2a and the rope is of length 4a. Let A be the area of the grass that the goat can graze. Prove that A ≤ 14πa² and determine the minimum value of A.

For the first part I proved it on the basis that the maximum area of grass the goat will have to graze will be when the goat is tethered to one of the corners of the barn, and I showed that this area is equal to 14πa² (I wasn't really sure how to explain why this is, it just seems obvious?)

For the second part I've said that the goat will have the smallest area to graze when it is tethered to the centre of one of the walls of the barn, and found the area it can graze to be  13πa². Can anyone verify that this is correct?

Thanks.

Last edited by Daniel123 (2008-02-22 10:49:01)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Goat

That's the answer I get. It'll be able to eat a semi-circle of grass with radius 4a in front of it, then on either side of the barn it'll eat a quarter-circle of radius 3a, and around the back of the barn it can eat a quarter-circle from either side, each with radius a.

So the overall area it can eat in this case is π/2(4²+3²+1²) = π/2(16+9+1) = 13π.

The question might want you to work out how much grass the goat can eat when it's tethered a distance x away from a corner of the barn though, and then differentiate and equate to 0 to find turning points.

It seems like it does, by its use of the inequality sign and asking you to find a minimum, but if it does then it's a really evil question.

Edit: OK, so if the goat is tethered ax away from one of the corners, then the amount of grass it can eat should be:

Multiply out and tidy to get a standard quadratic polynomial, differentiate to get something linear and easily equate to 0. Still though, evil question.

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It wanted to be normal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Goat

I thought it seemed a bit too straight forward. Nasty STEP I.

Thanks.

Last edited by Daniel123 (2008-02-22 10:51:33)