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?

Though this is basically algebra, the equations are particularly hard to set up.  To help myself along the way, I will include dimensional analysis.

There are four variables:
r=grass-growing rate=?pastures/day
c=cow's eating rate=?pastures/day
g=goat's eating rate=?pastures/day
s=sheep's eating rate=?pastures/day
Note that c, g, and s are all negative.

There are four equations:
c+g+r = -1 pasture / 45 days
c+s+r = -1 pasture / 60 days
c+r = -1 pasture / 90 days
g+s+r = -1 pasture / 90 days

Now for all that solving for the variables:
r = -1/90 pastures/day - c
c+g + (-1/90 pastures/day - c) = -1/45 pastures/day
g = -1/90 pastures/day
(-1/90 pastures/day) +s+r = -1/90 pastures/day
r = -s
c+s + (-s) = -1/60 pastures/day
c = -1/60 pastures/day
r = 1/180 pastures/day
s = -1/180 pastures/day

Now, to solve for x in the following equation:
c+g+s+r=-1/x pastures/day
x = -1/(-1/60 - 1/90 - 1/180 + 1/180) days
x = 1/(5/180) days
x = 36 days

There you have it--36 days.

Posted by Tristan on 2004-10-12 22:47:58