How Many Oxen Can Be Fed?

From: Anonymous
Date: Fri, 28 Jun 1996 9:22:58 -0400 (EDT) 
Subject: three pastures

 I don't understand this problem. Can you help me?

We have three pastures with grass of identical height, density, and 
growth rate. The first is three and one-third acres and can feed 12 
oxen for 4 weeks. The second is 10 acres and can feed 21 oxen for 
9 weeks. The third is 24 acres.

How many oxen can be fed on it for 18 weeks? 

Date: Fri, 28 Jun 1996 13:33:35 -0400 (EDT)
From: Dr. Anthony
Subject: three pastures

We must take account of growth of pasture while the cattle are 
feeding. Let e = extra growth per acre per week. So in the first 
situation: 

Total pasture = 3.5(1 + 4e) = k*12*4 
(k=constant of proportionality) 

In the second situation we have

Total pasture = 10(1 + 9e) = k*21*9

Dividing these expressions to eliminate k we get: 

3.5(1+4e)/{10(1+9e)} = 48/189

3.5(1+4e) = 2.54(1+9e)
1+4e = 0.725623(1+9e)
1+4e = 0.725623 + 6.5306e
0.274377 = 2.5306e

and so e = 0.1084237 and k = 10(1+9e)/189 = 0.10454 

Then in the last situation if x = number of oxen we have: 

0.10454*18x = 24(1+18e) = 24(1+18*0.1084237) = 70.839 

x = 70.839/(0.10454*18)

= 37.645 oxen - say 37 oxen

We could feed 37 oxen for 18 weeks on 24 acres. 

-Doctor Anthony, The Math Forum