A square mile contains 640 square acres, so a square acre is 1/640th of a square mile. There are 5,280 feet in a mile, so a square mile would be 27,878,400 square feet. Divide that by 640, or the acres in a square mile, and we get 43,560 square feet in a square acre. That’s a lot of steps.
If you’re an exercise enthusiast trying to rack up steps each day while trying to decide whether you want a riding lawnmower or the push kind, you might compare price with steps. Doing that starts with the first step – no, not with the mower! The first step is to figure out how we figure out how many steps it takes to mow one acre.
What Are The Elements Of The Problem?
We start with an acre. We are going to put a lawnmower of some kind on it. We are going to walk behind the mower as we push it. The lawnmower will cut a swath as wide as it can. We will walk as many steps as it takes to traverse the acre with those swaths.
What Are The Variables?
A variable is anything in a problem that won’t necessarily always remain the same. A constant is anything that does not change.
For example, an acre is 43,560 square feet. It stays the same. It is constant. The first variable in the problem is the lawnmower. How wide is the swath the mower cuts? Not all mowers have the same cutting width.
Is the grass an even height? Longer grass may take more than one pass, thus more steps. How many obstacles are there on the lawn? Rocks, trees, fence lines, brushy areas, creeks, and other things might reduce or increase the number of steps required. Is the landscape hilly? Strides uphill will tend to be shorter.
How tall are you? A person who is 6’2” will have a longer stride than someone who is 5’4”. The number of steps each of them needs to walk 100 feet will be different because the lengths of their legs are different.
Which Variables Are Relevant?
We are going to make some compromises to simplify matters. We will assume the grass is regularly mowed, and it is not uneven, thick, or difficult to cut. We will also assume the grass is a field with no obstructions, and it is relatively flat.
Then, the remaining variables seem to be the mower itself (how wide it is) and the height of the person mowing. All other things will remain constant for the purposes here, and the reader will be permitted to estimate how any constants will affect our final answer if they change.
Is This Leading Somewhere?
All of this information about variables and constants will enable us to write an equation that answers our original question under different circumstances.
However, we need to understand an acre in a different way. 43,560 square feet doesn’t help us as it is. However, if we find the square root of 43,560, we determine how many feet there are on each side of a square acre. That amounts to 208.71 feet long and 208.71 feet wide, or 2,504.52 inches long by 2,504.52 inches wide.
We also need to understand how a person’s height affects their stride. Mathematicians have discovered the length of a person’s step is equal to their height in inches multiplied by 0.413. Therefore, a 6 foot (72”) tall person would have steps equal to 72” x 0.413 or 29.736”, which is 2.48 feet.
Can We Put It Together, Now?
We have everything we need to write an equation that will tell us approximately how many steps it will take to mow one acre.
The first thing we need to do is take the width of the acre and divide it into rows the size of the mower’s cutting width. We will understand that as:
# rows = 2,504.52″/w, where w = the cutting width of the mower in inches
We will now multiply the # of rows by 208.71 feet to know how many feet will be walked back and forth across the rows in the field.
Oh, Oh. See The Error?
If you are like me, when you push a lawnmower back and forth across the lawn, you don’t line up the edge of each row exactly with the edge of the next. There is an overlap, right? We will say we overlap, liberally, about 4”. We now understand our number of rows to be:
2,504.52″/(w – 4″)
And the number of feet that walked as:
(2.504.52″/(w – 4″)) x 208.71’, where w = the cutting width of the mower in inches
The next step is to divide the number of feet walked by the length of each step to get a total number of steps. To do that, we take the height of our person in inches and multiply by 0.413. That number needs to be divided by 12 to express it in feet. We plug that in and bring it all together into:
[(2.504.52” ÷ (w – 4”)) x 208.71’] ÷ [(h x 0.413) ÷ 12] = x, where w = the cutting width of the mower in inches, h = the height of the person in inches, and x = number of steps to cut one acre,
Do You Want To Try It Out?
Cutting widths of push lawnmowers range from 14” for a reel mower to a range of 21” to 40” for self-propelled walk-behind mowers. Reel mowers take forever, so let’s start with a 21” mower pushed by a 6-foot-tall guy.
My high-school algebra teacher always made me show my work, but I’m not going to do that here. You check me out, instead (FYI, I round up the number of rows).
That person would take 12,466 steps. It would take 14,502 steps for a person 5’2”. Likewise, if the mower size increases to a 40” cutting width, the respective steps for each would be 5,896 and 6,859 steps.
By the way, I have to tell you. It takes me 43 steps to mow my one-acre lawn. If I am in the bedroom, it is 122 steps. That’s how far away the garage is. I have a riding lawnmower.