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### Feeding Oxen

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Date: 6/26/96 at 18:29:12
From: Anonymous
Subject: Pastures, Oxen Food

We have 3 pastures with grass of identical height, density and growth
rate. The first is 3 1/3 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 these pastures for 18 weeks?
```

```
Date: 6/26/96 at 21:48:44
From: Doctor Jodi
Subject: Memo: Pastures, Oxen Food

Hi there! Interesting question.  In order to answer it, we want to
calculate how much grass one ox eats in one week.  Does that make
sense?

So we'll say that 3 1/3 acres is 4*12 = 48 oxen-weeks (what 48 oxen
would eat in a week, or 1 ox in 48 weeks, etc...)

Now the question is, how many oxen-weeks are there per acre?  We can
find out by dividing:

This is 48 divided by 3 1/3 (which is the same as 10/3), that is to
say:

48 * 3/10 = 14.4 oxen-weeks per acre

Let's use the second set of numbers to verify this.

We also know that 10 acres is 21 * 9 = 189 oxen-weeks.
That would make 189/10 or 1.89 acres/oxen-week.
It looks like some other variables are involved.

IS THIS THE BEST APPROACH TO THIS PROBLEM?

-Doctor Jodi
The Math Forum
```

```
Date: 07/18/2001 at 23:42:50
From: David Medellin
Subject: Answer to the "feeding oxen" puzzle

Here is how to solve this puzzle. The answer is 36 oxen. Here's the
problem and the procedure.

We have three pastures with grass of identical height, density, and
growth rate. The first is 3 1/3 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 these pastures for 18
weeks?

The answer above does not include the growth rate in the equations.

Let k be the growth rate of the pastures. The amount of grass that
grows depends on the time and the size of the pasture, so
grass = k*area*time.

Let's find out how much grass grows in the first pasture in 4 weeks:

grass = k * (10/3) * 4,

but we have an initial amount of grass (grass_i = area * height). So
the TOTAL grass obtained in 4 weeks is:

grass = k(10/3)(4) + h(10/3)         (note that this is only
volume; since density is
constant we can work just
with volume)

Now we divide that equation by 48 to find how much grass an ox eats
in one week.

grass/(oxen*week) = (k(10/3)(4) + h(10/3))/48

We do the same with the second pasture and obtain a similar equation:

grass/(oxen*week) = (k(10)(9) + h(10))/189

We solve and get k = .9 and h = 10.8

and finally

(.9)(24)(18) + (10.8)(24) = 18*(number of oxen)

we solve for the number of oxen and get 36 - a very nice number.
```

```
Date: 07/19/2001 at 11:41:01
From: Doctor Rick
Subject: Re: Answer to the "feeding oxen" puzzle

Hi, David.

Good work! I agree with your solution. I'll just comment that, when
you substitute the values of k and h into your expressions for the
amount of grass eaten per ox-week, you get 1. That's because your
chosen unit for "amount of grass" is precisely the amount that an ox
is the only unit we have to work with, since we know nothing about
the actual height or density of the grass.

In other words, the precise definition of the variables is

h = initial density of grass in ox-weeks per acre
k = growth rate of grass in ox-weeks per acre per week

We are told that the first field feeds 12 oxen for 4 weeks, i.e. the
total amount of grass in the field, including that which grows over
those 4 weeks, is 48 ox-weeks. This amount can be calculated as
h times 3 1/3 acres plus k times 4 weeks times 3 1/3 acres. By this
approach we get the equations

(10/3)(h + 4k) = 48
10(h + 9k) = 189

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
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