Number of Times

[Guest]

A plow makes a path 18 inches in width. In order to plow half of a circular quarter section, how many times around it is needed ?

A quarter section is 1/4 of a square mile, or 160 acres.

Let T = number of times around.

How do I proceed ?

[Guest]

Please reply. Thanks.

[Guest]

Question..Is half of a circular quarter section a semicircle having the area of half a quarter section OR is it a circle having the area of half a circular quarter section. I assume the area or half a quarter section is 80 acres

If it is a circle of 80 acres then it is 80 x 4840 = 387200 sq yds.
Radius is SqRoot(387200 / Pi) = 351 yards.
Number of laps to plow to centre = 351 / 0.5 = 702 laps around the circle

[Guest]

"Question..Is half of a circular quarter section a semicircle having the area of half a quarter section OR is it a circle having the area of half a circular quarter section. I assume the area or half a quarter section is 80 acres" Not sure - the question is not worded clearly.


"Is half of a circular quarter section a semicircle having the area of half a quarter section" If so, would the answer be half of 702 ?

[Guest]

if a semicircle same area then sqroot(2 x 387200 / Pi) = 496 yards radius
number of laps = 496 / 2 / 0.5 = 496 laps of the semicircle

[Guest]

Thanks.

[Guest]

I have located a solution in another location:

160 acres = 160 x 160 = 25,600 square rods = area.

25,600 / 3.1416 = 8148.7 = square of radius.

The square root of 8148.7 = 90.27 = radius of field.

80 acres remaining if .5 is plowed.

80 acres = 12,800 square rods.

12,800 / 3.1416 = 4074.35 = r^2 of remaining.

Square root of 4074.35 = 63.8 = radius of remaining.

90.27 - 63. 8 = 26.4 rods = width of plowed ring.

26.4 rods = 5227.2 inches / 18 (width of plow) = 290.4 = number of laps.

I don't know why rods were used and I don't understand solution.

[Guest]

Just reading you posted solution.............

Looks like my assumptions were wrong and you did not keep me right

Looks like the circle is for 160 acres and 80 acres are to be plowed

So that gives a bigger circle and only plowing half of it.....
Now another question....which half.....
Is he plowing from centre out OR from outer edge in towards the centre?????

[Guest]

"Is he plowing from centre out OR from outer edge in towards the centre????? " The question does not state - please provide solution for both.

[Guest]

Earlier post about semicircle of 80 acres has same radius as circle of 160 acres, = 496 yards
My first solution of circle 80 acres had radius of 351 yards
Plowing the outer ring gives width of 496 - 351 = 145. Yards
Flowing 0.5 yards per lap means 290 laps. Same answer as your solution.
Flowing the inner circle 80 acres is same as my first solution

[Guest]

Thanks. Would you please explain the solution I provided ?

[Guest]

I have located a solution in another location:

160 acres = 160 x 160 = 25,600 square rods = area. This is the Ares 160 acres in square rods

25,600 / 3.1416 = 8148.7 = square of radius. This is the radius squared for 160 acres in rods

The square root of 8148.7 = 90.27 = radius of field. This is the radius in rods for 160 acres

80 acres remaining if .5 is plowed. This is the amount to be plowed

80 acres = 12,800 square rods. This it in square rods

12,800 / 3.1416 = 4074.35 = r^2 of remaining. The radius of the centre 80 acres...radius squared remaining

Square root of 4074.35 = 63.8 = radius of remaining. This is the radius of the centre 80 remaining

90.27 - 63. 8 = 26.4 rods = width of plowed ring. This is the difference between the outer edge of 160circle and the outer edge of the inner 80 circle expressed in rods

26.4 rods = 5227.2 inches / 18 (width of plow) = 290.4 = number of laps. This is it converted to inches and didided by the width of the plow to give the number of laps to plow the outer ring of 80 acres leaving the circle in the centre not plowed

I don't know why rods were used and I don't understand solution. This is worked out same as my solution except worked in rods and I worked in yards

[Guest]

Thanks again.

[Guest]

One question :

If I wanted to use feet for the measurements, would the computations be basically the same ?

[Guest]

Yes convert everything to feet and square feet and plow 1.5 feet each lap

[Guest]

Converting to feet:

80 x 43560 (square feet in acre) = 3484800

Radius = Sq, root (3484800 / pi) = 1053.2063 = 1053

1053 / 1.5 = 702 laps

The same as your answer. Where did I make the errors?

[Guest]

Converting to feet:

80 x 43560 (square feet in acre) = 3484800

Radius = Sq, root (3484800 / pi) = 1053.2063 = 1053

1053 / 1.5 = 702 laps

The same as your answer. Where did I make the errors? ..No errors, it is correct for 80 acre circle..........Yes the same as my answer for 80 acre circle and plowing it all...as was first assumed.

But then we discovered from your solution that the question was not about plowing ALL of a circle that had an "area equal to half a quarter" it was about plowing "half of a circle that had an area of a quarter" by plowing in laps around the outside towards the centre. When 80 acres of the 160 acre circle are plowed then 80 acres still remains as a circle in the centre.

[Guest]

If you want it as a formulae working in area of Acres and converting to feet.............

The take it as the difference of 2 concentric circles.........Plow the outer annulus (ring) and you are left with a smaller circle remaining.

The width of the ring will be the outer radius minus the inner radius. And take the area of the smaller remaining as half the larger total area.

So...width = .... Sqroot(2 x SmallArea x FeetPerAcre / Pi) Minus Sqroot(SmallArea x FeetPerAcre / Pi) in feet

Gives.... Sqroot(2 x SmallArea x FeetPerAcre / Pi) x ((Sqroot(2)) -1) feet

Gives.... Sqroot(2 x SmallArea x FeetPerAcre / Pi) x (0.414) feet

Gives... Sqroot(2 x 80 x 43560 / 3.142) x (0.414) feet

Gives......... 1053 x 0.414 = 435.94 feet....this is width of ring plowed

Each lap plows 1.5 feet.....So...... 435.94 / 1.5 = 290.6 laps to plow the outer ring.

[Guest]

Thanks again.

[Guest]

Using the relationship between the linear dimensions of objects and their areas.

If two objects have a ratio in area of 4 to 1 (4:1) then their linear dimensions (length or breadth) will be in the ratio of (sqrt4 to 1) or (2:1).

If two objects have a ratio in area of 2 to 1 (2:1) then their linear dimensions (length or breadth) will be in the ratio of (sqrt2 to 1).

Original area is 160 acres of which 80 acres are to be plowed leaving 80 remaining. So ratio of original area to remaining area is a ratio of 2:1
So length of the radius of each circle will be in ratio of (sqrt2 to 1)......also any other corresponding linear dimension on each circle.

Area...remaining in sqfeet.... 80 x 43560 (square feet in acre) = 3484800 sq feet

Radius = Sq, root (3484800 / pi) = 1053.2063 = 1053 feet .... this is radius remaining

Original area is two times the area remaiming so original radius is (Sqroot(2) x 1053)

Original radius is .... 1.414 x 1053 = 1488.9 feet

width of ring is the difference...... 1489 - 1053 = 436 feet width = 436 / 1.5 = 290.6 laps.