# Thread: Finding Area Function in Terms of Length

1. ## Finding Area Function in Terms of Length

My math teacher has only gone over questions like these once in class and does not really explain clearly, so I’m struggling with the whole question. Here it is. “There is 1500ft of fencing that needs to be closed off at all sides. Find the area function in terms of length, dimensions, and maximum area that can be enclosed.” (I also need to state the domain and range after but I know how to do that). Could someone be so kind as to explain how to do this question? Thank you!

2. ## Re: Finding Area Function in Terms of Length

Originally Posted by universityletmein
My math teacher has only gone over questions like these once in class and does not really explain clearly, so I’m struggling with the whole question. Here it is. “There is 1500ft of fencing that needs to be closed off at all sides. Find the area function in terms of length, dimensions, and maximum area that can be enclosed.” (I also need to state the domain and range after but I know how to do that). Could someone be so kind as to explain how to do this question? Thank you!
$\text{Area}=l\cdot w$. Now you know that $2l+2w=1500$ so $w=750-l$

Thus $\text{Area}=l(750-l)=750l-l^2$. Graph that and determine the Max.

3. ## Re: Finding Area Function in Terms of Length

I have a problem with the fact that this does not say what shape the fence encloses. It does say "closed off on all sides" so I guess we can assume this is a rectangular field. The area of a rectangle is, as Plato says, length times width. Then, as the perimeter is 1500, the area is $l(750- l)= 750l- l^2$. That is quadratic and can be written $-(l^2- 750l)$. You can determine the maximum by "completing the square".

However, a circular fence, with circumference 1500, would give a larger area than any rectangle.

4. ## Re: Finding Area Function in Terms of Length

Originally Posted by HallsofIvy
I have a problem with the fact that this does not say what shape the fence encloses. a circular fence, with circumference 1500, would give a larger area than any rectangle.
What have I missed all my life?
I never knew that a circle could have sides, under the usual understanding of 'side'.

5. ## Re: Finding Area Function in Terms of Length

Max area is a square (and a square IS a rectangle)
so 1500/4 = 375, making area 375^2 = 140625 sq.ft.

6. ## Re: Finding Area Function in Terms of Length

Yes, Plato, I did say, and you deleted it from your copy, that "It does say "closed off on all sides" so I guess we can assume this is a rectangular field."

7. ## Re: Finding Area Function in Terms of Length

The problem statement is faulty, because it omits stating that the area is a
rectangular shape.

The first part of the problem statement should be closer to this:

"There is 1500 feet of fencing that is needed to close off all sides of a rectangular area."

8. ## Re: Finding Area Function in Terms of Length

In fact, I thought that the wording "There is 1500ft of fencing that needs to be closed off at all sides" was poor wording. It isn't the fencing that needs to be "closed off at all sides"! It is the region enclosed by the fencing.

9. ## Re: Finding Area Function in Terms of Length

It’s my bad that I forgot to mention that there is a picture along with the question that shows the area as a rectangle.

10. ## Re: Finding Area Function in Terms of Length

Well then, it's definitely a square (a square is a special rectangle).