Farmer Joe owns a cow, a goat, and a sheep. The animals each eat grass at a constant rate, and the grass grows at a constant rate. And Farmer Joe occasionally lets them eat the grass on a small pasture of his.

• If the cow and the goat graze together, the pasture is bare after 45 days.
• If the cow and the sheep graze together, the pasture is bare after 60 days.
• If the cow grazes alone, the pasture is bare after 90 days.
• If the goat and the sheep graze together, the pasture is bare after 90 days, also.

How long will it take for the pasture to be bare if all three animals graze together?

OK, so I did this in my head and I got 36 days.  My analysis completely ignores the 4th statement.  Haven't read the others yet.

Here is my reasoning.  First, I imagined a single blade of grass.  The cow eats, in one day, the blade's daily growth plus 1/90 of the initial length of the blade on Day 1.  or g+1/90
Cow eats:   g + 1/90
Cow + Goat eats:   g+1/45 =  Cow plus 1/90
Cow + Sheep eats:   g+1/60 =  Cow plus 1/180

Cow eats:  g+1/90
Goat eats:  1/90
Sheep eats: 1/180

All three eat:  g+1/90 + 1/90 + 1/180 = g + 1/36
So the grass is gone in 36 days.

Now adding in the information from the 4th equation tells us that grass grows at the same rate as the sheep eats, ie 1/180 th of the length of the blade of grass on day 1.   So if the sheep grazed alone, the grass would last forever.

So why were the cattle ranchers and sheep ranchers always having gunfights in the old westerns?

Posted by Larry on 2004-10-13 04:00:46