On a certain pasture grass grows at a even rate. It is known that 40 cows can graze on it for 40 days before the grass is exhausted, but 30 cows can graze there as long as 60 days.

How many days would pasture last if 20 cows were to graze on it ?

Use the formula distance = rate x time.

Let the distance (height of the grass) be measured in units in which the current height of the grass is 1. Let the rate at which a cow eats the grass be c and the rate at which grass grows r. Then, as distance = rate * time,

1=(40c-r)*40
1=(30c-r)*60

Respectively these are equivalent to
1600c-40r=1
1800c-60r=1

Solving, 1200c = 1, or
c=1/1200
then
r=1/120

If d is the number of days the pasture would last if 20 cows were to graze on it, the formula becomes
1=(20/1200)d
or
1=(1/60 - 1/120)d = d/120
so d = 120 days, the answer.

As an aside, we can see that 10 cows grazing would exactly balance the growth rate of the grass, so 10 is a sustainable number of cows with no time limit.

But again, 120 is the answer to the question posed.

Posted by Charlie on 2003-08-13 14:46:16