Feeding OxenDate: 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 eats in a week. There is nothing circular about this definition; it 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 Your solution follows. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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