Number - Acres - Solution/Explanation

[Guest]

In a tract of land there are as many acres as there are boards in the fence surrounding the tract. The boards are 11 feet in length and the fence is 4 boards in height. Determine number of acres.

Solution provided with statement:

Side ^2 / 160 = Number of acres in the tract, using rods.

4 x 16.5 x side = Perimeter in field.

4 x (4 x 16.5 x side) / 11 = Number of boards in fence.

Side^2 / 160 = 4(4 x 16.5 x side) / 11 = 24 x side.

Side^2 = 160 x 24 x side = 3840 x side.

Side = 3840 rods = 12 mi.

3840^2 / 160 = 92160 acres.

Please explain the solution.

[Guest]

You seem to have got the same answer as myself, but not sure why you did some of the conversions.
The boards length was given in feet and area required in acres so don't know why you worked in rods.???
I think I have less and simpler calculations.........
...........
Boards are horizontal as one rail above each other, each 11 feet in length. Take square area of land as you did.

Let B = boards in 1 rail along 1 side. So 4 rails on the fence.

So length of side is 11B.......Total boards = 4 x 4 x B = 16B boards......and 43560 sq.feet in 1 acre

Area if square = 11B x 11B / 43560 = 16B
121B / 43560 = 16
121B = 16 x 43560

B = 16 x 43560 / 121 = 5760 boards....Total boards = 16 x 5760 = 92160 boards = 92160 acres

[Guest]

I didn't work out the solution. It was included with the problem. I am in agreement with you on the feet, acres, and rods.

[Guest]

Another solution included:

16 = Number of acres between 2 panels of fence on opposite sides of field.

Acre = 43560 sq. ft.

16 acres = 16 x 43560 sq. ft. = 696960 sq. ft.

11 ft. = width between 2 panels.

12 mi. = 63360 ft. = 696960 / 11 = width of field.

144 sq. mi. = 12^2 = area of field.

Sq. mi. = 640 acres.

144 sq. mi. = 144 x 640 = 92160 acres.

[Guest]

Looks like this is a solution from a very basic forum user and not from any reliable mathematical source...??????
I fail to see the logic in it.....except you can explain...................

Another solution included:

16 = Number of acres between 2 panels of fence on opposite sides of field.???????where does this come from, why do we need it????

Acre = 43560 sq. ft.......This is just stating a fact.....conversion

16 acres = 16 x 43560 sq. ft. = 696960 sq. ft.........Irrelavent....just the number of sqft in 16 acres....why????

11 ft. = width between 2 panels. ......where did you find that out...11 ft is length of 1 board.?????

12 mi. = 63360 ft. = 696960 / 11 = width of field........Why work out 12 miles....just guessing?????

144 sq. mi. = 12^2 = area of field. ......how do you know the field is 12 miles square....?????

Sq. mi. = 640 acres........just a conversion again

144 sq. mi. = 144 x 640 = 92160 acres........How did you even think this might be the answer..........

[Guest]

I didn't work out the second solution either. It was included with problem.

[Guest]

Did you get it in a published text book or an a forum or webpage..??

[Guest]

I got it in an old archived book.

[Guest]

Why not post the full text of your archive solution so that we can try to understand it.
OR Provide you own reasoning of how it was calculated. I cannot see what logic is being followed........

The question is simple .......so below I will give another method for solving. I don't think it is any more simple that my first solution but this is it to add another dimension..??........and I will fully document it so anyone should be able to understand it.

Let us start from a per unit basis and then ratio up to the real problem..........

Take 1 acre = 43560 sq.feet......A Square of sides = 208.7 feet by 208.7 feet

Number of boards for 1 rail along 1 side will be 208.7 / 11 = 18.974 and there are 4 rails so 4 x 18.974 = 75.895 boards.
And there are 4 sides so 4 x 75.895 = 303.579 total boards around 1 square acre. Let the area of the large tract be = (A^2) acres so then the sides are sq.root of that equal to A ......

According to the question for the tract, the number of boards 303.579 x A = A^2

So 303.579 = A .......

So......Total number of boards for tract = 303.579 x 303.579 = 92160 = this is also the number of acres.

.................As I said.......another Simple solution...................

[Guest]

The posting was the question and solutions. I can also post their explanation.

[Guest]

Yes, go ahead and post the explanation, I would be interested to see it, Thanks.....
Here is another solution for good measure...............

Another way to look at it is to start with the big problem and bring it down to size...........

There are 4 rails made up of boards 11 feet long and 4 sides to the tract of land.
So that is equivalent to 16 rails each the length of 1 side of the land.
Let the area of the land be " A^2 ", then the length of 1 side will be " A ".

Total Number of boards 16A / 11 = A^2/43560 = the area of land in Acres

16A x 43560 / 11 = A^2

16 x 43560 / 11 = A

63360 = A = length of 1 side of the tract of land in feet.....

Area of land = 63360 x 63360 = 4014489600 sq.feet = 4014489600 / 43560 = 92160 acres

And total number of boards = 16 x 63360 / 11 = 92160

.............. its still a Simple solution ..........................

[Guest]

Explanation with problem:

For every board in fence, an acre of land in tract for 4 boards, or one panel 4 acres. A panel on the opposite side of field would be 4 acres. A tract 11 ft. in width and 8 acres would be between 2 panels on opposite sides of field. The boards on the other 2 sides have as much value as the first 2 sides. So twice the area of rectangle between 2 opposite panels for the area between 2 opposite panels in the total tract. There are 16 acres between 2 opposite panels in the tract. The length of rectangle is 16 x 43560 / 11 = 63360 = 12 mi., length of side of tract.

The length of a side, in miles, can be determined by multiplying the number of boards in height of fence by 33 and by dividing the product by length of a board in feet. This is for any problem of this type.

[Guest]

Comments on............Explanation with problem:

Quote "For every board in fence, an acre of land in tract for 4 boards, or one panel 4 acres" ..............

Yes, OK, One board represents one acre. 4 boards in a panel represents 4 acres, but, does not enclose 4 acres. It would take 4 panels formed as a square to enclose 16 acres. This is same as 16 boards encloses 16 acres, and equals 1 board per acre.

Let "B" = number of boards surrounding the whole tract of land. and that also equals the number of acres. Let "A" equal the number of acres.

Therefore ..... B = A ...... but we need to write "A" in terms of "B" to only have 1 unknown and be able to solve.

So lets get the length of a side in terms of "B".
"
"B" is all the boards, so "B" divided by 4 boards will be used on each side. The boards are in "panels" "4 rails" or "4 boards" high....one "rail" above each other. So each "rail" running the length of a side uses "B divided by 4" and divided by 4 again boards = (B / 16) boards, and each 11 foot long, placed end to end along the side of the tract of land. So the length of the side of the tract is (11 x B / 16) feet. And it is square so the width is the same and area is (11 x B / 16) ^2.

So the area "A" is (121 x B^2 / 256) sq.feet. And that equals ((121 x B^2 / 256) / 43560) Acres. .....and that equals the total number of boards.
So .... B = ((121 x B^2 / 256) / 43560)

OR 1 = ((121 x B / 256) / 43560)

So.......... 43560 = (121 x B / 256)

So ..........43560 x 256 / 121 = B

So B = 92160 ... and this equals "A" the area in acres. OK, Yes, I think that all follows through, and that is the sum done.

I don't follow the reasoning outlined in the Quoted parts of the explanation below...........

Quote, "A panel on the opposite side of field would be 4 acres."

Yes, OK, A panel comprising 4 boards represents 4 acres if the number of boards equals the number of acres.

Quote "A tract 11 ft. in width and 8 acres would be between 2 panels on opposite sides of field."

Yes, OK, There would be an area of land 8 acres representing "the 8 boards" between 2 panels at oposite sides of the field. I don't follow the logic that it would be 11 feet wide.??. Two panels cannot enclose any land, there would also be boards along the sides. I think they are actually also using the fact that 8 boards equals 8 acres and the length of the boards are 11 feet long and go around the perimeter. But nowhere in the explanation is there any reasoning to show a relationship between the area and the perimeter.

Quote "The boards on the other 2 sides have as much value as the first 2 sides. So twice the area of rectangle between 2 opposite panels for the area between 2 opposite panels in the total tract. There are 16 acres between 2 opposite panels in the tract."

Yes, OK, If you do the same the other way you will get a similar situation. I am not sure what "the other way means". Are they doing langth x width?? or just adding another long rectangle 11 feet wide??.

Quote "The length of rectangle is 16 x 43560 / 11 = 63360 = 12 mi., length of side of tract.

??...I am not sure how this calculation follows on from the above reasoning....??...It is simply a calculation that says that 16 boards which equals 16 acres converted to square feet divided by the length of each board gives the length of a rectangle that is 11 feet wide....which I queried earlier.
This actually works out to be the length of the side of the tract in feet, but how can one see this from the reasoning given.

Quote "The length of a side, in miles, can be determined by multiplying the number of boards in height of fence by 33 and by dividing the product by length of a board in feet. This is for any problem of this type."
Yes, it seems to work on this occasion but no explanation of how it is derived.

[Guest]

I don't understand your per unit or big problem solution.

[Guest]

Reply.....I don't understand your per unit or big problem solution.........?????

It was supposed to be based on the fact that the lengths of the sides of a square is the sq.root of its area
and if the the square sides are say doubled in length then the area is 4 times the original.

Like this simpler example.........
Take a square field 1 acre in size, its sides will be Sq.root of 1 in length and say it takes 10 boards on each side for the fence.
That is 40 boards to fence an acre fully all around. Question is how many boards does it take to fence a larger field if the number of boards is equal to the number of acres.

Let the larger field be "A^2" acres (that is "A^2" times 1 acre) then sides will be A times the length that 1 acre sides were so will be "A".

The number of boards needed will be 40A and that equals A^2

So 40A = A^2 .....or.....A = 40

Number of boards is 40 x 40 = 1600 ....and ......Acres is 40^2 = 1600......so both are equal


In the problem you query we didnt know how many boards to fence an acre to start with.

So took sq.root of 43560 = 208.7 feet long and boards 11 feet long so 18.97 boards for 1 rail of boards on one side of an acre.

And 4 rails and 4 sides gives 16 x 18.97 = 303.579 boards to fully fence 1 acre.

Let the area of the large tract be = (A^2) acres so then the sides are sq.root of that so equal to A ......

According to the question for the tract, the number of boards 303.579 x A = A^2

So 303.579 = A .......by dividing across by A

So......Total number of boards for tract = 303.579 x 303.579 = 92160 = this is also the number of acres.

[Guest]

And I suppose I should also say for the simpler example the large field sides will now be 40 time bigger and need 10 x 40 boards = 400 boards along each side to fully fence so for around the whole field will be 4 x 400 = 1600 which we got when we multiplied 40 x A = 40 x 40 = 1600

[Guest]

I have read and read your explanation but I still don't understand.

[Guest]

What bit do you not understand
Do you not understand the simpler example either

[Guest]

I don't understand:

"It was supposed to be based on the fact that the lengths of the sides of a square is the sq.root of its area
and if the the square sides are say doubled in length then the area is 4 times the original."

"Let the larger field be "A^2" acres (that is "A^2" times 1 acre) then sides will be A times the length that 1 acre sides were so will be "A"."

"According to the question for the tract, the number of boards 303.579 x A = A^2"

"Total Number of boards 16A / 11 = A^2/43560 = the area of land in Acres

16A x 43560 / 11 = A^2

16 x 43560 / 11 = A"

[Guest]

Is it the sums (as in the numbers) you don't understand or is it the reasoning (how side length and area are related) that you do not understand.

If a square has side length "A" feet then its area is "A^2" or A x A sq.feet. That is the basis of any area calculation.

So if a square has sides 2 feet x 2 feet then its area is 2 x 2 = 4 sq.feet.

If I double the length of the sides of the square, the the sides will be 4 feet long and the area will be 4 x 4 = 16 sq.feet
So I doubled ( by 2... = 2 x 2) the sides and got 4 times ( 4 x 4 )the original area.
So the ratio of ....Sides to Area is 2 to 4 for the small square and 4 to 16 for the large square
For small square 4 is 2^2 and for the large square 16 is 4^2......So if sides were "A" the area would be "A^2" sot the ratio of Sides to area would be "A" to "A^2" for the square of size "A".

And for a unit square of sides equal to 1, then the area is 1 x 1 = 1^2 = 1 also. so we can say that the sides of the square sized "A" would be "A" times bigger than the unit square and the area of the square sized "A" would be "A^2" times bigger than the unit square.

And because the sides are "A" times bigger then the number of boards required to go around the perimeter fence will be "A" times bigger. And the area will be "A^2" bigger and the question say this will equal the actual number of boards if expressed in acres. And it is divided by 43560 to convert sq.feet to acres.