A rancher wants to build a fence adjacent to a river. He has 12 miles of fencing. What dimensions should he use to maximize the area of the fence? Note that the rancher does not need to fence next to the river. Show all work.

I took college algebra years ago, and I don't remember how "college algebra" would relate to this in terms of functions.

I would simply use a 3 x 3 square, and that would give you 9 square miles of area inside that square, but I might not be understanding these details correctly. Will someone help me on this? This is for my little sister, and she has been through a lot. I could just tell her to go to a tutor, but she has been through hell since the start of this year. I'll do what I can to help her from 1400 miles away, but i'm of no use to her when it comes to word questions like this.

Thanks in advance.

the geometric figure with the shortest circumferences for any given area is a circle which leads to the suspicion that a semi-circle bordering on the river might be the right solution
for a fence with 90 degrees corners the square would give the maximum area which at the river would lead to a 4 x 4 square since one side can be replaced
that would lead to an area of 16 sqm but if you use the river as the diagonal of a square you would get a bigger area
which shows that the river side needs to be taken (care)fully into account, so it needs a little analysis

is what you quoted the complete question or are there some boundary conditions?


This post was edited by brmv on Feb 16 2014 08:03pm

That is the exact question. Nothing else is included

"A rancher wants to build a fence adjacent to a river. He has 12 miles of fencing. What dimensions should he use to maximize the area of the fence? Note that the rancher does not need to fence next to the river. Show all work."

I appreciate your help.

I'm assuming you want it to be a rectangle.
Since you want to build it adjacent to a river, that means you only need fencing for 3 sides of the rectangle.
You are given the total length of fence is 12 miles. Now you need to build an equation that represents the area of a rectangle.

A = L * W is the generic one

between the 3 sides of the rectangle, the total length will be 12.
Let the sides perpendicular to the river be x. That means the side across from the river is (12 - 2x).

Now that we have our L and W, plug in all your values:

A = x(12 - 2x)

simplify:

A = -2x^2 + 12x

without using calculus, to find the minimum or maximum, you use the vertex.

vertex x-value: x = -b/2a
x = -12/(2)(-2) = 3

So we know the perpendicular sides are length 3

the side across from the river is (12 - 2x) = (12 - 2(3)) = 6

So the side across from the river is length 6

in that case i stick to my suspicion that the paddock will be a semicircle
don't have the time atm to go any further
but since 'Azrad' is here you are in good hands
he has better tools available than i have

AHHH!! My stupidity!

Note that the rancher does not need to fence next to the river. <--I thought they were just adding this for the heck of it. The wording threw me off for some reason. THANK YOU SOO MUCH!!!

I truly appreciate your help. It's hard to show how thankful I am, but be assured that I feel VERY grateful. Thanks, and have a wonderful day.

this is a classic question. I believe the answer TritonV8 gave is the one they are looking for, but I also believe the semi-circle is actually the correct answer.

the perimeter of a semi-circle is pi*r (not counting the river)

pi*r = 12
r = 12/pi

the area of a semi-circle is:

(1/2)*pi*r^2, so

A = (1/2)*pi*(12/pi)^2 = 72/pi ≈ 22.9 miles^2

The rectangle is 6x3 so its area is 18 miles^2

Of course this does not prove the semi-circle is best, but it beats the rectangle. But again I have to stress, I'm betting the answer that is "wanted" is the rectangle, and the person who wrote the question forgot to put that constraint on the problem. Heh, kind of leaves you in a sticky situation...

Azrad, I appreciate your reply. I think you're absolutely correct. Thank you for your response.

Thank you everyone!

Does the river have bends that adjunction the property line?

Is the area assumed flat or are there different levels of terrain?

"Note that the rancher does not need to fence next to the river." -With this you should assume that the Rancher has an infinite amount of property to build this fence on as the question does not specify otherwise. Go for the maximum amount of square miles that could be registered under these ideal circumstances.

which is the same area you get when you consider the river to be the diagonal in a square with sides of 6 miles each, as 6^2/2 = 18
but yes, the question is how far the sister of the op is in the class - will it allow solutions which are not rectangular

and @ 'FLegend' by the wording of the question is can be assumed that the river is running straight and the terrain is level

This post was edited by brmv on Feb 16 2014 09:10pm