Interesting Math Problem

November 12, 2013

 

Suppose you have a grassy field, and cows eat grass at a constant rate.                 
Keep in mind, the grass keeps growing continuously.

48 cows can clear all the grass off the field in 90 days.

120 cows can clear all the grass off the field in 30 days.

How many cows would be needed to clear all of the grass in 16 days?
Round up to the nearest whole cow.



A cow eats a certain amount of grass in one day, call it c.
The field grows by a certain amount each day, call it g.

The field has some initial amount of grass: i

i + 90 g - 48*90 c = 0
i + 30 g - 30*120 c = 0

1.2 i + 36 g - 4320 c = 0
i + 90 g - 4320 c = 0

0.2 i - 54 g = 0
0.2 i = 54 g
i = 270 g

The field starts with 270 days' growth.

i + 30 g - 30*120 c = 0
300 g - 3600 c = 0
g = 12c

i + 90 g - 4320 c = 0
360 g - 4320 c = 0
g = 12c ... confirmed

It takes 12 cows to eat one day's growth in one day.

So 270 initial + 16 days' growth = 286 days' growth

In 16 days, n cows eat 16n / 12 days' growth.
16n/12 = 286
16n = 3432
n = 214.5 cows
Round to 215
215/12 = about 18 days' growth consumed each day
18 * 16 = 288 which exceeds the 286.

 

 

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