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?

(In reply to Non-linear Approach by bernie)

If we let r be defined as the initial proportional eating rate for an animal, and V(t) to be the volume of grass in proportion to the initial grass, then:

V(t)=((a+t)/a)-r(a+t)ln[(a+t)/a]

When V(t)=0 then ar=1/ln[(a+t)/a]

So, 4 equations, 4unknowns:

ac=1/ln[(90+a)/a]

ag+as=1/ln[(90+a)/a]

ac+ag=1/ln[(45+a)/a]

ac+as=1/lm[(60+a)/a]

The first two tell us that c=g+s.  The sum of the last two is three times the first one:

1/ln[(45+a)/a] + 1/ln[(60+a)/a] = 3/ln[(90+a)/a]

I cannot solve this nicely for a, but a quick excel "solver" provides a=69.83057...

We want the t that makes V(t)=0 when r is 2c (2c=c+s+g).  This would mean that 2ac=1/ln[(a+t)/a] where c is (1/a)/ln[(90+a)/a] giving t=a(((90+a)/a)^.5-1).

t is then 35.81535...

I don't think the answer is going to work out to exactly 36 days. 

Posted by bernie on 2004-10-25 13:09:02