Optimizing Pasture Fencing: Min. Fencing Length & Area

[domyy]

Homework Statement

A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough gas for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Should I use 2y + x = 180,000 ?? Minimize A(x) = (x)(y)
I've also tried
4y + 4x = 180,000;
4y + 3x = 180,000;
4y + 2x = 180,000

I don't know what is going to work here.

 

[Ray Vickson]

Homework Statement

A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough gas for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Should I use 2y + x = 180,000 ?? Minimize A(x) = (x)(y)
I've also tried
4y + 4x = 180,000;
4y + 3x = 180,000;
4y + 2x = 180,000

I don't know what is going to work here.

Don't guess; work it out carefully.

First: always draw a picture.

Now, assuming you have done that: what is x? What is y?

In terms of x and y, what is the length of fencing used? What is the area enclosed? Now, what are the problem conditions telling you?

In problems of this type you should NOT try to jump right away to the final formulation; you should proceed step-by-step.

 

[domyy]

x = L
y = W

I spent hours, at first, trying to calculate x for [ 4y +( 2x or 3x or 4x) = 180,000]. I thought 180,000 corresponded to the size of the rectangular farm and that I had to calculate the fence. Then, I reread it and realized 180,000 corresponded to the fence.

Now, since no fencing is needed along the river, I assume there will be only one x:

2y + x = 180,000

However, none of the equations I've mentioned so far lead me to the final answer. I've tried and I keep getting y = 45,000 or y=180,000.

 

[haruspex]

I keep getting y = 45,000 or y=180,000.

What do you consider to be wrong with that?

 

[domyy]

My book says the answer is 600m x 300m. I was expecting to get to a point where I would find 600m and 300m individually.

I don't know what else is going to work here besides the equations I've already tried. I am probably not being able to visualize it.

 

[haruspex]

Ah - ok. I presumed your equations were sort of reasonable, but they're not. You are not told the length of fencing is 180,000. You are told the area to be enclosed is 180,000. What is an expression for that in terms of x and y?

 

[domyy]

Thank you so much!
When you said "you are told the area to be enclosed is 180,000" made me think and i finally realized how to solve it.

Here it is:

p = 2y + x

a = (x)(y) = 180,000

now, from here i didn't have any problem. My issue is always with building the equations. Thanks !