41
A
1
A
2
=L
1
L
2
2
.
Note that it doesn’t matter where we choose to measure the linear size, L, of
an object. In the case of the violins, for instance, it could have been mea-
sured vertically, horizontally, diagonally, or even from the bottom of the left
f-hole to the middle of the right f-hole. We just have to measure it in a
consistent way on each violin. Since all the parts are assumed to shrink or
expand in the same manner, the ratio L
1
/L
2
is independent of the choice of
measurement.
It is also important to realize that it is completely unnecessary to have a
formula for the area of a violin. It is only possible to derive simple formulas
for the areas of certain shapes like circles, rectangles, triangles and so on, but
that is no impediment to the type of reasoning we are using.
Sometimes it is inconvenient to write all the equations in terms of
ratios, especially when more than two objects are being compared. A more
compact way of rewriting the previous equation is
A
.
L
2
.
The symbol “
.
” means “is proportional to.” Scientists and engineers often
speak about such relationships verbally using the phrases “scales like” or
“goes like,” for instance “area goes like length squared.”
All of the above reasoning works just as well in the case of volume.
Volume goes like length cubed:
V
.
L
3
.
If different objects are made of the same material with the same density,
.
=m/V, then their masses, m=
.
V, are proportional to L
3
, and so are their
weights. (The symbol for density is
.
, the lower-case Greek letter “rho”.)
An important point is that all of the above reasoning about scaling only
applies to objects that are the same shape. For instance, a piece of paper is
larger than a pencil, but has a much greater surface-to-volume ratio.
One of the first things I learned as a teacher was that students were not
very original about their mistakes. Every group of students tends to come
up with the same goofs as the previous class. The following are some
examples of correct and incorrect reasoning about proportionality.
Section 1.2Scaling of Area and Volume