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?

For the problem to work as if its water into a leaky bowl, it means that the progressively smaller patch of grass produces prortionately more and more growth.

Two ways to look at this problem are:

1.  Animals eat a little bit of every blade of grass uniformly, and the blades get longer by growing a set length per unit of time.

2.  Animals eat whole blades, or leave ground bare going in from the edges of the field so that the eaten grass areas cannot grow again.  The remaining blades grow a set length per unit of time.

Both assumptions are extreme; a "real life" situation would be somewhere in the middle, but I think the second scenario is much closer to "real life."  Imagine an animal eating a few cells off of every blade of grass over a whole field in some simultaneous fashion versus an animal eating grass by pulling it up by the roots, area by area, eating its way across the field.

Leaky bowl solves under the first assumption.  You need more math for the second one.

Posted by bernie on 2004-10-27 21:30:02