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### How Many Oxen Can Be Fed?

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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?
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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

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