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 ?

Assumptions: The amount a cow eats in a day does not alter by cow or day. The grass stalk grows at a continuous rate and amount (no limits on grass height). The base unit of amount is the amount of grass a cow eats in one day.

A governing identity is:
Current Grass Amount = (Grass Growth Rate - Cow Eat Rate)*Days + Initial Grass Amount

The problem gives us two specific instances of this formula:

0 = (Grass Growth Rate - 40)*40 + Initial Grass Amount
0 = (Grass Growth Rate - 20)*60 + Initial Grass Amount

This system gives a Grass Growth Rate of 10 units per day and an Initial Grass Amount of 1200 units. Substituting back into the identity, we can answer several questions: 20 cows can graze for 120 days, 10 or less cows can graze indefinitely, and 1210 or more cows will last a day or less.

Edited on July 14, 2005, 3:19 pm

Posted by owl on 2005-07-14 15:16:49