Proportions and Scaling
Name: Bill S.
Let us say that I live in a very uniform neighborhood
where all the lawns are the same size, the ground is level, etc. Let us
also say that it takes me 1 hour to mow any of the lawns. To mow 2 lawns
would take 2 hours (assuming I do not need to take any breaks). In
general form it takes N hours to mow N laws.
You could also express the time in minutes by saying that it takes 60
minutes to mow one lawn, 120 minutes to mow two lawns, etc.
I want to know how long it will take to mow N^2 lawns. One might think it
would take N^2 hours. That seems simple enough, does it not? If we choose
N = 3, we find that it takes 3^2 or 9 hours to mow 9 lawns.
If we use minutes for the computation we obtain very different results. 3
hours = 180 minutes. 180^2 = 32,400 minutes. That works out to 540 hours!!!
Why can't you "square" time? ;-)
The formula is NOT (time)-squared. The formula is (time per lawn)
multiplied by (N-squared), where N^2 is the number of lawns. If T=total
time, R=rate (or time per lawn), and N^2=number of lawns, then T=R*N^2. In
the first case, you were squaring the number of lawns, but not then
multiplying by 1hr per lawn. You got the correct number because multiplying
by one doesn't change the number. The units were incorrect in the first
version. N^2 is "lawns", not "hours". You must then multiply by your rate
of 1hour/lawn to get a time unit. Multiplying by 60minutes/lawn gives you
the time in minutes, and it is the same amount of time.
Dr. Ken Mellendorf
Illinois Central College
According to your specification it requires 1 hr to mow 1 lawn i.e.
(1h/lawn) * N lawns = N hours.
It makes no difference HOW you arrive at the N number of lawns. For example,
if the lawns were in a developing subdivision one might find that as the
year progressed and new houses were completed, a new lawn was added
according to the empirical formula: N = (M) + 0.25(M^2) - 0.1(M^3) where
M=1 for January, M=2 for February, M=3 for March and so on. Rounding to the
nearest lawn number, N, it still requires 1 hr for each lawn regardless of
how the number of lawns was arrived at.
This is a matter of units. When you square a number, you also square the
units. So if you take 3 lawns and square the value, you get 9
lawns^2. Likewise when you square the hours, you get 9 hours^2. If you
use 180 minutes and square that value, you get 32,400 minutes^2. Now to
convert square minutes to square hours you must do the math to cancel out
the other unit (Hopefully this will come through in the e-mail)
32,400 minutes^2 1 hour 1 hour
* --------- * ---------- = 9 hours^2
60 minutes 60 minutes
See, you can square time, you just have to watch the units when doing
so. The way you were suggesting, the answer of 540 had the units of 540
hour-minutes. Not a unit I am familiar with.
Hope this helped.
Chris Murphy, PE
Mechanical Design Engineer
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Update: June 2012