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?

I will call the statements S1 – S4. I will call the rate of cow, goat, and sheep consumption c, g, and s, respectively. I don’t know if this is right, but I will say that the pasture starts out with fixed amount of grass P, and then the rate of grass grown is r.

S1: 45c + 45g = P + 45r          c+g = P/45 + r
S2: 60c + 60s = P + 60r          c+s = P/60 + r
S3: 90c = P + 90r                   c = P/90 + r
S4: 90g + 90s = P + 90r          g+s = P/90 + r

Add S3 and S4 together, see what happens. c+g+s = P/45 + 2r Hmmm, that doesn’t work. We need just 1r.

What about add S1 and S2, then subtract S3?
S1 + S2 = (c+g) + (c+s) = (P/45 + r) + (P/60 + r)
2c+g+s = P/20 + 2r

Then subtract S3 from that and get
(2c+g+s) – c = (P/20 + 2r) – (P/90 + r)
c+g+s = P*(7/180) + r

So the number of days is 180/7 which approximately equals 25.7 days.

?????

Posted by nikki on 2004-10-12 14:17:00