September 13, 2004
Problem of the Month

The Lawnmower Problem



We bought a new lawnmower this year.   When I asked my wife how the first mowing went, she replied that it seemed to go faster.   I answered, "that's because our new mower cuts a 22 inch wide path while our old mower only cut a 21 inch path."

So the question is: How much faster can she mow the lawn given the following assumptions?

  1. Assume that our lawn is a rectangle 150' x 105'.
  2. Assume that the lawn will be mowed in strips and always in the same directions (either back and forth along the width or back and forth along the length).
  3. My wife can mow at the rate of 120 feet/min.
  4. It takes 5 seconds to turn around at the end of each strip.




Solution to the Problem:

If you mow along the longer side, it will take 160 seconds or 2.65 minutes less to mow the lawn.
If you mow along the shorter side, it will take 230 seconds or 3.83 minutes less to mow the lawn.

To figure this out, 150 feet equals 1800 inches.
Dividing by 21 inches means that it will take 85.71 or 86 paths (mowing along the shorter sides).
Dividing 1800 inches by 22 inches requires 81.81 or 82 paths.
Dividing 105 feet or 1260 inches by 21 inches means that it will take 60 paths to mow the lawn. Dividing 1260 inches by 22 inches means that it will require 57.27 or 58 paths.
The number of times required to turn around will be one less than the number of paths.
Here is a comparison of the data for each mower:

Mower Mow along
width or length 		# of paths # of times to
turn around 		Time to mow each strip Total Time
old 21" length 60 59 1.25 min or 75 sec 4,795 sec or 79.9 min
new 22" length 58 57 1.25 min or 75 sec 4,635 sec or 77.3 min
old 21" width 86 85 .875 min or 52.5 sec 4,940 sec or 82.3 min
new 22" width 82 81 .875 min or 52.5 sec 4,710 sec or 78.5 min




Correctly solved by:

1. Walt Arrison Philadelphia, Pennsylvania
2. Jeffrey Gaither Winchester, Virginia
3. Chris McCormick Virginia Tech,
Blacksburg, Virginia
4. James Alarie University of Michigan -- Flint,
Flint, Michigan
5. John Funk Ventura, California
6. Ben Schultz Winchester, Virginia
7. S. McCann ----------
8. Julie W. Dallas, Texas
9. Emily Auerbach Columbus, Georgia
10. Tristan Collins Winchester, Virgina
11. Mandy Auerbach Columbus, Georgia
12. Bobby Amos Winchester, Virgina
13. David Amos Winchester, Virgina
14. Kevin Jacobson Columbus, Georgia
15. Jonathan Jansson Tullängskolan, Örebro, Sweden
16. Joakim Berggren Tullängskolan, Örebro, Sweden
17. Viktor Logrell Tullängskolan, Örebro, Sweden
18. Erik Ekman Tullängskolan, Örebro, Sweden


Send any comments or questions to: David Pleacher