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?
Well, I seen a few of the solutions, and can say I'm most impressed by Larry's soltuion. I couldn't see the picture that clearly trying to solve it on my own, so I had to go through my own method.
I let these be my variables
I = Initial bales of grass in field.
R= Growth Rate in bales per day.
C= Cow's consumption rate in bales per day.
G= Goat's consumption rate in bales per day.
S=Sheep's consumption rate in bales per day.
Then we have...
1.) 45(C+G) = I + 45R
2.) 60(C+S) = I + 60R
3.) 90C = I + 90R
4.) 90(G+S) = I +90R
Immediately we can see that C= G+S from 3 and 4, but we still have 4 equations with 5 unknowns. The way I got around this was just let R = 1 bale/day and just see what everything came out after we had 4 equations with 4 unknowns. The new equations with R= 1 are...
1.) I = 45(C+G) - 45
2.) I= 60(C+S) - 60
3.) I= 90C-90
4.) I=90(G+S) - 90
Now we can solve and see that
S=1, G=2, C=3, and I = 180 when R=1.
So (C + G + S)X = I + RX whereas X is what we're after and we can see that X= 36 days.
This problem is interesting in that we don't know exactly what we have initially in the pasture or the growth rate or the cow consumption rate etc., and still don't know even when we have found the solution. However, we do know the relationships between those variables, and that is enough to answer the question in the problem. A pretty neat problem SilverKnight! :)
Answer In Bales In Lieu of Blades
