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        riddles >> medium >> Newton's Cows.
        
(Message started by: rloginunix on Dec 24^th, 2013, 8:19pm)

Title: Newton's Cows.
Post by rloginunix on Dec 24^th, 2013, 8:19pm

This problem has been allegedly authored by none other than Isaac Newton himself.

The daily vertical growth rate and the per square area density of the grass on a grazing field are constant. 70 cows consume the entire grass on the entire field in 24 days while 30 cows do it in 60.

How many cows does it take to consume the entire grass on the entire field in 96 days?

Hint: read. the. problem. carefully.

Note: this problem is a follow up for the "Painting the Wall" thread.

Title: Re: Newton's Cows.
Post by jollytall on Dec 25^th, 2013, 10:36pm

I get [hide]20[/hide].

Title: Re: Newton's Cows.
Post by jollytall on Dec 25^th, 2013, 10:46pm

And since there typically comes the question about the how:

How not to:
There are far too many variables if you approach it incorrectly. The field is x1 m2, the grass originally x2 m long, grows at a rate of x3 m/s, a cow eats x4 m/s grass, while the density is x5 1/m2. 5 unknowns, 2 equations. Will not work. You can even further complicate it, since we do not know if the grass originally all have the same length.

How to:
[hide]Every cow eats a bit of every grass every day. We killed the area, the density. Also let the bit (or bite) they eat every day the unit.
Also change to some non SI units.
So now we have only two unknowns, y1 bite long initial grass length in average and y2 bite/day growth rate.
Now we know:
30*60=1800 bites equals to y1 + 60*y2 and
70*24=1680 bites equals to y1 + 24*y2.
From here it is elementary.[/hide]

Title: Re: Newton's Cows.
Post by rloginunix on Dec 26^th, 2013, 10:52am

Excellent job, jollytall. Excellent job. Your answer is correct and your logic is flawless.

At first I've used a similar approach myself but then after looking at the numbers Newton used for days more closely a worm of a vague hunch started turning in my tummy. Some more digging and playing with numbers yielded yet another solution. Not too different but I think is worth mentioning. I'll post it shortly. Meanwhile may be someone else can do it too.

Title: Re: Newton's Cows.
Post by rloginunix on Dec 26^th, 2013, 11:13am

Good remark on approaching it incorrectly too, jollytall. Here's a quick anecdote underscoring one.

Two politicians are having a lunch break after an agricultural bill passing session:
- Those Depublicans and Remocrats have some nerve! How many cows!
- Calm down, Billy. I just love how you got up, looked them straight in the eye and said that 24 is 4 times less than 96 and hence it would take 4 times less cows!
- 4 times less cows.
- That's right, Billy. 70 over 4 is 17.5 cows.
- 17.5 cows.
- That's right.
- And I loved how you, Johnny, got up, looked them straight in the eye and said that 60 is 1.6 times less than 96 and hence it would take 1.6 times less cows!
- 1.6 times less cows.
- That's right, Johnny. 30 over 1.6 is 18.75 cows.
- 18.75 cows.

Title: Re: Newton's Cows.
Post by rloginunix on Dec 26^th, 2013, 3:29pm

I noticed that the [hide]distance between 24 and 60 is 36. The same distance as between 60 and 96. In other words between the 1-st and the 2-nd and the 2-nd and the 3-rd experiments the grass gets to grow the same number of days, 36.

We can have either 3 fields acted upon in parallel or 1 field acted upon in sequence. Either way the initial amount of grass on the field(s) is always the same. And for these particular numbers we don't need it.

The 70-cow herd needs to make 70x24 = 1680 bites to clear the field. The 30-cow herd needs to make 30x60 = 1800 bites to do the same. That's 1800 - 1600 = 120 extra bites over 36 extra days. The same extra 36 days during which the X-cow herd needs to make the same extra 120 bites which is 1800 + 120 = 1920 since the grass grows at the same rate:

X * 96 = 1920

from where it follows that X is 20.

Of course we've got lucky with the numbers here. If they were different we would[/hide] do it jollytall's way.

Title: Re: Newton's Cows.
Post by rmsgrey on Jan 6^th, 2014, 8:19am

It's not luck when a published problem has numbers that just happen to give an easy solution...