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?

This problem is analogous to pouring (grass growing) water into a bowl at a constant rate and the bowl is leaking (animals eating grass) at a constant rate.  Assuming that the animals start grazing on the grass when the grass covers the pasture (i.e., the bowl is full), let the variables below be defined as folloows.

v_c = the cow's speed of grazing

v_g = the goat's speed of grazing

v_s = the shee's speed of grazing

v_gr = the grass' speed of growing

Then the conditions can be translated into math expressions as follows:

1 - v_c*(45) -v_g*(45) + v_gr*(45) = 0

1 - v_c*(60) - v_s*(60) + v_gr*(60) = 0

1 - v_c*(90) + v_gr*(90) = 0

1 - v_g*(90) - v_s*(90) + v_gr*(90) = 0

1 -v_c*(t) - v_g*(t) - v_s*(t) + v_gr*(t) = 0

Then from the system of first four linear equations, we get

v_g = 1/90, v_s = 1/180, v_c = 1/60, v_gr = 1/180

So "t" in the fifth equation is t = 36.

It takes 36 days for the grass to become bare.

Posted by doremi on 2004-10-27 13:38:27