Solving the Sheep and Grass Puzzle: Maximum Number of Sheep

[mb34]

I got this question from my teacher to be completed by the beginning of Dec. It goes like this.

sheep eat grass at a fixed rate
grass grows at a fixed rate

1 sheep in a pen eats grass forever
8 sheep eat grass in 5 days then starve
9 sheep eat grass in 4 days then starve

I figure from the 8 sheep that 20% is eaten each day so by day five all the grass is gone and no new new grass is available. This led me to 1 sheep eating 2.5% a day. By day 40 all grass would be gone except enough grass has grown to replace eaten grass. I think that 12.5% is replaced every 5 days (this is my calculation).

Can anyone help me?

The question is what is the maximum amount of sheep that can be kept in a pen and eat grass forever?

 

[BobG]

I got this question from my teacher to be completed by the beginning of Dec. It goes like this.
sheep eat grass at a fixed rate
grass grows at a fixed rate
1 sheep in a pen eats grass forever
8 sheep eat grass in 5 days then starve
9 sheep eat grass in 4 days then starve
I figure from the 8 sheep that 20% is eaten each day so by day five all the grass is gone and no new new grass is available. This led me to 1 sheep eating 2.5% a day. By day 40 all grass would be gone except enough grass has grown to replace eaten grass. I think that 12.5% is replaced every 5 days (this is my calculation).
Can anyone help me?
The question is what is the maximum amount of sheep that can be kept in a pen and eat grass forever?

The sheep eat more than 20% per day. Not only have they eaten all the grass that existed in the pen, but they've also eaten all the grass that has grown in 5 days, or 9 days, or whatever.

You'll need to set up two simultaneous equations with two variables (what percentage of the pen's grass does each sheep eats per day; what percentage of the grass regrows each day). You started with one pen of grass. Keep in mind that you don't have just 8 sheep for 5 days or 9 sheep for 4 days - instead you have sheep-days (kind of like calculating man-hours for labor).

 

[Danger]

Sheep?
Did someone mention sheep?

 

[ranger]

Sheep?
Did someone mention sheep?

You wouldn't

 

[mb34]

sheep problem coupled with no math classes for 10 years problem.

that's the part I don't get. If I knew how much grass regrew each day I could solve this easy enough, but i just don't see how to arrive there. I also see that in 5 days 8 sheep eat all of one pen and any regrowth. That means (i think) that they must eat at least 2.5% of the pen each day. but that's 2.5% of THAT pen. If it were a bigger pen 2.5% of that pen would be a larger amount consumed by each sheep and the question states sheep eat a constant amount. So how do I come up with two equations 1 for sheep-days and 1 for regrowth of grass per day without more information?

I see from a mod post that you do not do the homework for me, but any equation help or just plain extra guidance would be appreciated. It's been 10 years since my last math class and while I was never totally clueless there were definitely times when I thought, HUH?!?

By the way I did work this out somewhat and I came to the conclusion that the most I could keep in the pen forever was 1. using what I think I derived from the equation 2 sheep would eat all the grass and regrowth in 39 days whereas the 1 sheep eats the first pen amount in 40 days but enough was regrown in that same time to sustain what was eaten. Or my answer basically was 2.5% a day eaten and 2.5% regrown each day.

sorry to go long. please help it must be obvious by now I do not know my head from my ass. thank you all in advance.

 

[BobG]

Given two equations (which you are), you can sovle for both x and y.

In other words, if I had two equations (real easy, just to illustrate the example):

$$2x + 4y = 16$$
$$4x - 3y = -1$$

Multiply the first equation by -2 to get:

$$-4x - 8y = -32$$
$$4x - 3y = -1$$

Then add the two equations together to get:

$$0x - 11y = -33$$

Solve for the remaining variable and substitute back into either original equation to solve for the variable you eliminated.

Using that method, it's just a matter of setting up your problem to fit the method.

The initial amount of grass is 1 - one pen of grass. The pen is the same size, the initial amount of grass in the pen is the same for all scenarios.

The pen can support up to four sheep forever, not one.

You have to solve this using simultaneous equations. Let x be the percentage of grass that one sheep can eat per day. Let y be the amount of grass that grows each day.

You have a set number of days before the amount of grass is reduced from 1 to 0. That's the number of days that the grass grew. You have a certain number of sheep times the number of days of grass eating.

1 + (sheep*days)x + (days)y = 0