Question

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.

Answer 1

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.

??

Answer 2

yeah...

Answer 3

no contradiction. the grass keeps growing.

\[initial + (grassgrowthrate * days) = (numcows * eatingrate) \]

for simplification

x + y * z = i * j

in the context of this problem
x + 30y = 120j
x + 90y = 48j

Answer 4

what is initial?

Answer 5

the amount of grass that the field had initially

Answer 6

yeah i figured out that.

Answer 7

so you set them equal to each other because you assumed they ate all the grass

Answer 8

I didn't assume, they said that they "cleared the field"

Answer 9

errr, that was given

Answer 10

so what exactly are you equating , what is your left side and right side

Answer 11

total grass grown = total grass eaten (over the days )

Answer 12

the initial grass plus the growth of the grass is equal to the grass that the cows ate in other words,

when you subtract the total grass minus the grass eaten, it should equal zero, thus my equation.

Answer 13

ok but thats 2 equations in 3 unknowns

Answer 14

the grass is growing while they are eating

Answer 15

grass growth must be some function of days

Answer 16

i started this way, 1 cow eats an ammount of grass each day is m
the grass grows by the amount n each day and if the initial is a
then we get a+90n=48*90m
a+30n=120*30 m

Answer 17

well you can subtract equation 1 from equation 2, that eliminates x the initial

Answer 18

by eliminating the initial i will find the relation between m and n

Answer 19

how did you get grass growth rate is 90 and 30 ?

Answer 20

nevermind

Answer 21

that was days

Answer 22

Yeah sritama I was actually just about to correct my equations to add the days on both sides.

Answer 23

yeah,i took the grass growth rate n/each day ... so it must b 90n and 30n

Answer 24

oh ok gingerkid 101

Answer 25

ginger can you redo your equation

Answer 26

i like your equation, its very logical :)

Answer 27

x + 30y = 120 * 30 * j
x + 90y = 48 * 90 * j

where x is in grass,
y is in grass per day
j is in grass per cow per day

Answer 28

yeah thats a little odd, grass per cow per day

Answer 29

how does that come out in units, grass/ (cow/day) or

Answer 30

probably it means the amount of grass taken by each cow per day

Answer 31

grass/(cow * day) or grass/cow/day
it's all the same.

Answer 32

in other words, if you have a certain number of cows, you can multiply it by that number to get 30 cows * 1 grass/cowday = 30 grass/day is resulting from having 30 cows.

Answer 33

anyway the equations fully played out are...

x + 30y = 3600j
x + 90y = 4320j

that's what I'm at right now.

Answer 34

ok now its linear algebra, one moment

Answer 35

x + 30y - 3600j=0
x + 90y- 4320j=0

Answer 36

I get x = 3240 j
y = 12j
j = j

Answer 37

actually we know we want 16 , so we have another equation

Answer 38

x + 30y = 3600j
x + 90y = 4320j
x + 16 y = 16 i * j

Answer 39

However, we also know that 8 additional cows are needed to balance the original proportion, which can possibly give us a 3rd equation?

8j - 60y = 0
or else the proportion would be even.

therefore, we now have a 3 part system

8j - 60y = 0
x + 30y - 3600j=0
x + 90y- 4320j=0

Answer 40

how did you get 8 additional cows?

Answer 41

yeah,thats my question too

Answer 42

8 additional to 1/3 of 120. maybe I'm just talking out of my retricenow haha

Answer 43

I don't even know. meh, I never had to do anything like this and I took up to calc 3 haha

Answer 44

well this is a solid start

Answer 45

i got 12j = y , so 12 cows it takes to eat the grass grown in 1 day