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Example: Cost of transporting tomatoes
Question: Roughly what percentage of the price of a tomato comes
from the cost of transporting it in a truck.
The following incorrect solution illustrates one of the main ways you can go
wrong in order-of-magnitude estimates.
Incorrect solution: Let’s say the trucker needs to make a $400
profit on the trip. Taking into account her benefits, the cost of gas,
and maintenance and payments on the truck, let’s say the total cost
is more like $2000. I’d guess about 5000 tomatoes would fit in the
back of the truck, so the extra cost per tomato is 40 cents. That
means the cost of transporting one tomato is comparable to the cost
of the tomato itself. Transportation really adds a lot to the cost of
produce, I guess.
The problem is that the human brain is not very good at estimating
area or volume, so it turns out the estimate of 5000 tomatoes fitting in the
truck is way off. That’s why people have a hard time at those contests where
you are supposed to estimate the number of jellybeans in a big jar. Another
example is that most people think their families use about 10 gallons of
water per day, but in reality the average is about 300 gallons per day. When
estimating area or volume, you are much better off estimating linear
dimensions, and computing volume from the linear dimensions. Here’s a
better solution:
Better solution: As in the previous solution, say the cost of the trip
is $2000. The dimensions of the bin are probably 4 m x 2 m x 1 m,
for a volume of 8 m
3
. Since the whole thing is just an order-of-
magnitude estimate, let’s round that off to the nearest power of ten,
10 m
3
. The shape of a tomato is complicated, and I don’t know any
formula for the volume of a tomato shape, but since this is just an
estimate, let’s pretend that a tomato is a cube, 0.05 m x 0.05 m x
0.05, for a volume of 1.25x10
-4
m
3
. Since this is just a rough
estimate, let’s round that to 10
-4
m
3
. We can find the total number
of tomatoes by dividing the volume of the bin by the volume of one
tomato: 10 m
3
/ 10
-4
m
3
= 10
5
tomatoes. The transportation cost
per tomato is $2000/10
5
tomatoes=$0.02/tomato. That means that
transportation really doesn’t contribute very much to the cost of a
tomato.
Approximating the shape of a tomato as a cube is an example of another
general strategy for making order-of-magnitude estimates. A similar situa-
tion would occur if you were trying to estimate how many m
2
of leather
could be produced from a herd of ten thousand cattle. There is no point in
trying to take into account the shape of the cows’ bodies. A reasonable plan
of attack might be to consider a spherical cow. Probably a cow has roughly
the same surface area as a sphere with a radius of about 1 m, which would
be 4
p
(1 m)
2
. Using the well-known facts that pi equals three, and four
times three equals about ten, we can guess that a cow has a surface area of
about 10 m
2
, so the herd as a whole might yield 10
5
m
2
of leather.
Chapter 1Scaling and Order-of-Magnitude Estimates
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