A man has an acreage where grass grows continuously and evenly. If 20 cows eat all the grass in 96 days and 30 cows eat all of the grass in 60 days, then how long does it take 70 cows to eat all of the grass?
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Cows eating Grass10 guesses to Cows eating Grass |
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April 27th, 2010 | Category: 
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is it 96 days or 90 days for 20 cows?
it should take 26 days
Carole, you are very close but that is not the answer. And yes, it is 96 days.
there is infinite solutions, since the product of the two given data pairs are different we can choose “any” function(s) to describe the cows eating process.
hyperbola with x=0 asymptote:
days(cows) = 2160*1/cows – 12
days(20)=96; days(30)=60;
days(70)=~18.86 -> 19 days
hyperbola with y=0 asymptote (probably you are looking for this function):
days(cowd) = 1600*1/(cows-10/3)
70-> 24 days
24 days is correct!!!
Here’s how the Dude figured it:
Let’s say one cow can eat X pounds of grass per day. The field grows Y pounds of grass per day. At the start, the field already has N pounds of grass available. At the end, the field has 0 pounds of grass remaining. So we set up an equation for the amount of grass for the two cases, adding growing grass and subtracting eaten grass.
For 20 cows eating for 96 days: N + 96Y – 20x96X =0
For 30 cows eating for 60 days: N + 60Y – 30×60 X =0
Or: N + 96Y – 1920X = 0 and N + 60Y – 1800X = 0
When you subtract the two equations, you get:
36Y – 120X = 0, or 36Y = 120X or 3Y = 10X
In other words, in one day ten cows can eat as much as the field can grow in three days.
Now, substitute this back into either of the two original equations to find out how much grass was in the field to begin with.
N + 60Y – 1800X = N + 60Y Y = 0 and N = 480Y
Now we set up the equation for 70 cows eating and the field growing for T days, until the grass is gone.
N + TY – 70 TX = 0, or 480Y + TY – 21TY = 0 480Y, TY = 0
Hence, T – 24
So the 70 cows finish off the field in 24 days.
Great job, Det. You’re today’s winner!
96? Not very sure but could be right
24 days
24
25/7
Lets find the grass growth rate.
Condition 1 ==> 20 x 96 = 1920
Condition 2 ==> 30 x 60 = 1800
So rate of grass growing in 36 days (96-60) is 120 (1920-1800)
So daily grass growing rate = 120/36
Initial grass 1600 (1920 – (96 * 120/36) or 1800-(60 * 120/36))
Let say N is number of days
Then
Initial Grass + N * Rate = Number of Cows x N
1600 + (120/36)N = 70N
70N – (120/36)N = 1600
200N = 1600 x 3
N = 4800 / 200
=24 days