Number of Acres: Solution / Ezplanation

[Guest]

In a field, the number of boards in the fence enclosing the field equals the number of acres in the field. The boards are 11 ft. in length and 4 boards in height. Determine number of acres.


Solution provided with problem:

Let x = one side of the square in ft.

x^2 / 43560 = number of acres.

16x = number of feet of boards in fence.

16x / 11 = number of boards in fence.

x^2 / 43560 = 16x / 11

x = 63,360

x^2 / 43560 = 92,160 acres or 144 sections.



I know there are 43,560 sq. ft. in an acre.

Please explain the solution.

[Guest]

Solution provided with problem: ......explained

Let x = one side of the square in ft.

x^2 / 43560 = number of acres. .......Question say this equals the number of boards

16x = number of feet of boards in fence. .....the fence is 4 boards high so 4 x number of feet each side and field has 4 sides so 16x

16x / 11 = number of boards in fence. ....each board is 11 feet so how many 11 feet.....and this equals the number of acres

x^2 / 43560 = 16x / 11 ......so put acres on (LHS) and boards on (RHS) of equation and solve...

Divide both sides by "x" gives...

x / 43560 = 16/11

x = 16/11*43560 = 63360

x = 63,360 = length in feet of the side of the square field.

x^2 / 43560 = 92,160 acres or 144 sections.

[Guest]

Thanks.

[Guest]

In your mathematical opinion, what is the reasoning with such an unusual problem ? ( Not just using algebra to solve).

[Guest]

It can also be looked at as a "Ratio and Proportion" problem.
Work out boards for 1 acre, then ratio up the to the actual area.

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

Number of boards for 1 board high along 1 side will be 208.7 / 11 = 18.974 and there are 4 boards high so 4 x 18.974 = 75.895 boards along 1 side.
And there are 4 sides so 4 x 75.895 = 303.579 total boards around 1 square acre.

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

Linear lengths of sides will be proportional to number of acres, so if Area is A^2, side is A and number of boards is 303.579 x A

According to the question, the number of boards equals the number of acres, so...... 303.579 x A = A^2

So ..... divide across by A...gives..... 303.579 = A .......

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

[Guest]

Ok, thanks again.