Grazing Cows
Alas the answers suggested by BMAD and plasmid were both incorrect.
Let a = initial amount of grass, in kg say
b = quantity of growth, in kg per day
c = quantity consumed by one cow, in kg per day
k = number of cows and
N = number of days it takes to consume all available grass
Now we can write:
initial quantity of grass + growth over N days = quantity consumed by k cows in N days,
a + bN = kcN
Making k the subject we can determine the relationship between k and N:
k = a/c (1/N) + b/c where a, b and c are constants.
Thus the relationship is a hyperbola with asymptotes N = 0 and k = b/a.
The diagram above shows a hyperbola passing through the two fixed points, (6 cows, 3 days)
and (3 cows, 7 days), but there will be a family of such curves, depending on the ratio b/a, the
quantity of growth per day to the original quantity.
As for the question of how many days it would take one cow to consume the total feed, if we let
k = 1 and solve for N we obtain N = a /(c -b)
Thus the time taken for one cow to eat all the feed depends on a, the original quantity of feed, and (c-b), the difference between c, the quantity eaten per day, and b, the growth per day.
If c > b N will be some finite number of days, if c ≤ b the cow will never catch up with the growth.