# Thread: Area as a function of width

1. ## Area as a function of width

Hey guys, I'm holly. I was wondering if anyone would help with a problem. How do you express the area of a rectangle as a function of its width? There is an example in my book where they show it but that doesn't help. The example is something like you have 2000 feet of fence and you want to make four sheep pens. Each pen has to be a rectangle of the same shape.

Love ya

2. ## Re: Area as a function of width

Hi Holly, let's make the example a little easier to helpl you understand.

Let's say we only had one sheep pen fenced by 500 feet. Calling the length $L$ and the width $W$ then it makes sense that the total perimeter of the fence is $2L+2W = 500$ .

We know the area of a rectangle is $A = L \times W$ so looking back at the first equation and solving for $L$ we get

$2L+2W = 500$

$2L = 500 - 2W$

$L = 250 - W$

Substituting this back into the area equation for $L$ we get $A = (250 - W) W$

Do you follow?

3. ## Re: Area as a function of width

Thank you so much pickslides for replying. You were looking at one fence with the perimeter of 500. In the example of the book they were looking at 4 with a perimeter of 2000. So would it be 4L + 4W = 2000 ? Because I'm looking at 4 rectangles?

4. ## Re: Area as a function of width

I think it would be $8L+8W=2000$

5. ## Re: Area as a function of width

So I still end with L=(250-w)w ?

Yep.

7. ## Re: Area as a function of width

If you have 4 pens, exactly the same shape, with all fences separate:
_ _ _ _
|_| |_| |_| |_|
there are 8 vertical fences and 8 horizontal so yes, that is the correct answer.

But you could also place the pens in a large rectangle partern so that each shares a fence with the other.
_ _
|_|_|
|_|_|

Taking W for the horizontal dimension and H for the vertical, there are 6 horizontal fences and 6 vertical fences so we have 6W+ 6H= 2000.

8. ## Re: Area as a function of width

Hey HallsofIvy thanks for replying. Does the fact that the farmer wants to produce four sheep pens placed together in a grid matter? It does right? Lol. Thanks for making that little diagram thingy HallsofIvy because with the grid we have 6W+6H=2000. You guys are the best

9. ## Re: Area as a function of width

So 6L + 6W = 2000
Solving for L I get (2000/6) - W
Now to express the total area of the pens as a function of the width of each pen:

A = 2L X 2W

Now substitute L with (2000/6) - W

A= 2(2000/6 - W) X 2W.

Is this right?

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### area function in terms of the width

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