Math Problem.?

Question

So heres the math problem..

Theres a farmer and he wants to get some cattle to increase his monthly profit. The trouble is, is that he has no room to put them. So he goes out and buys 2000 ft. of fence. He has a barn which is 80 ft wide and 160 ft long, however it is completely full of machinary and NO cattle can go in the barn. The question is, what way can he put the fence to get the MOST square feet , so he can put the most cattle in , and make the most money. Remember the cattle must fit in the fencing and they cant go in the barn however you could use 160 ft of the side of the barn as fencing. You can put the fencing in any shape but I think it is going to have to go into a square. thanxs :)

Answer 1

We're gonna use one of the 160 ft long sides of the barn that isn't an entrance as fencing. The 160 ft will not be an entire side, but will be used as the middle of one of the sides of the fencing. So now, we essentially have 2160 feet of fence.

Now, I'm gonna use Calculus on this one, if you haven't gotten there yet, you can ignore it.

We'll call the width of the fence x and the length y
So 2x + 2y = 2160
We are trying to optimize the area of x*y, so
A = xy

Let's put our amount of fence equation into terms of x.
2x = 2160 - 2y
x = 1080-y
now we'll plug that back into the area formula
A = 1080y - y^2
take the derivative: A' = 1080 - 2y
There is a zero for A' at y=540, showing that the area of the fence is increasing from values of 0 to 540, and starts decreasing after 540. Therefore, we can say safely that the optimum length of fencing is 540 feet.
Now, plug the 540 back into the amount of fencing equation to get your width. 540*2 + 2x = 2160
2x = 1080
x = 540. So we want the width of the fence to be 540 also, so the fence is a square.

so the answer is that he can start with a 160 ft. wall of the barn, and extend the wall with fencing on either side so that the length of that wall is now 540 feet, so he'd need 380 feet of fence for that wall. For all the other walls, he's gonna need the full 540 feet of fencing.

540 + 540 + 540 + 380 = 2000 ft. of fence, and 291600 square feet of plot.

Yes, a circle would be the best way to optimize area as compared to diameter, but I don't believe circular fencing was intended to be an option in the problem, mainly because fencing is bought in straight "lines". Unless this farmer is a master welder to, and can efficiently weld the straight fencing into a circle, which isn't feasible, our square is the only option.

I'm going to guarantee that, although a circle would yield higher area, that it is incorrect, due to the fencing ordeal, and the square is the intended answer for the problem.

XoXoNatitaXoXo, good thought. I like your answer. Let's not make it so hard on the farmer so that he has to cut his fence into 25 pieces though. :-D. I probably wouldn't have come up with that as a new 6th grader, and I just took Calculus last year in 10th grade.

Answer 2

Let x+160 = length
Then [2000- (2x +160)]/2 = width = 920-x
So A = (x+160)(920-x) = -x^2 +760x + 147200
dA/dx = -2x +760
Set dA/dx = 0 getting x = 380
So you are right it's a square with sides = 540 feet.
This includes 160 feet of the barn as part of the fence.
Total area = 291,600 feet
Total perimeter = 4*540 = 2160 = 2000 feet of fence + 160 feet of barn wall.

Answer 3

Well, ok. Im not that smart yet because Im barely in 6th just got out of 5th grade but well I tried to and figured this out. Ok well if you have a shape of 25 sides, you can make each side 80 cm and that gives the cattle space and he can have the machines there. If you cant do 25 sided shape just do about 333 hexagons. ( 2000 divided by 6 is 333.33) So obviously it can't be a square because a 25 sided figure isn't going to ba a quadrilateral. Hope this helps and well I just did this 4 fun so thanks for the math problem! I don't really mind if Im wrong, I just hope 2 Be right!

Answer 4

Actually you would get the most square footage by approximating a circle. If you have 2000 ft of fence plus 160 feet of straight wall you have 2160 feet to use as the circumference of your "circle".

Circumference is equal to pi * d, so d = 2160/3.14 = 688ft.
The largest possible area would be a circle with a diameter of 688 feet.

This provides a square footage of just under 371,500 ft.

Answer 5

Actually a circle is the figure bounded by the smallest perimeter as compared to area. Can the fence be circular or include a curved area? If so, the problem will be far more difficult.