Answer to the September 13, 2004 Problem of the Week

The Lawnmower Problem

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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.

 

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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	Number of paths	Number 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