Some of the word problems in this class deal with the effects of
*scaling*. Scaling an object means multiplying every linear
dimension of it by the same factor. Thus you **change the size**
of the object, **but not its shape**. The laws of scaling described
on this page explain why there are no giants or dragons, why animals
can't fly if they weigh more than 50 pounds or so, why a spider can
jump out of an airplane without getting hurt while an elephant would
be likely to break some bones if he jumped off a table, why stars are
spherical, why potatoes cook quicker if you cut them into pieces, why
kids get cold more easily than adults, why basket ball players weigh
so much, why the largest animals (blue whales) that ever lived live in
water, and a host of other every day phenomena. Moreover, analyzing
the effects of scaling is an excellent exercise in using radicals
and powers
.

While of course most objects and creatures have complicated
shapes, everything about scaling can be understood in terms of
*cubes*. You can think of any other object as having been
composed of many small cubes. When you scale the object, whatever
happens to each of the small cubes also happens to the object as a
whole. So if you understand cubes you understand almost everything
about scaling!

As indicated on the left in this Figure, consider a cube that's
one inch long, one inch wide, and one inch high. The area of each of
its six square faces is one square inch, and since it has six faces
its total (surface) area is six *square inches*. The volume of
the cube is one *cubic inch*. That's where the words
*square* in *square inch* and *cubic* in
*cubic inch* come from!

If we multiply the length, width, and height of that cube with two,
say, we obtain a new cube, shown on the right in the Figure. It's
bigger, but it's a cube, it has the same *shape* as the
original, smaller, cube. As the Figure shows, we can think of that new
cube as having been obtained by piling up eight of the old cubes in
two layers of four cubes each. The volume of the larger cube,
therefore, is *eight* cubic inches, eight times the volume of
the smaller cube. The area of each face of the new and larger cube is
four square inches, and so its total area is
square inches, four times the area of the original cube. In building
the larger cube, we multiplied the length of the cube with , its area with , and its volume with .

More generally consider a cube of
length . Its volume equals .
Notice the exponent *3*.
This is why we use the phrase *cubed* to express a third power.
It's what we need to compute the volume of a cube.

Similarly, each of the six square faces of cube has an area of . That's why we use the word *squared* to denote a second power, we need it to compute the area of a square.
The *total* area of the cube is
. Now consider multiplying the length (and width and height)
of that cube with a factor . We obtain a new cube of length,
width, and height . Its volume is

This is true not just for cubes. We could do a similar analysis for a sphere of radius . Its volume equals and its area equals Note the factors and . As before, if we multiply with a factor we multiply the volume with and the area with .

This is true in great generality. As mentioned above, **just think of
a general object as being approximated by many small cubes, and think
about what happens to each of these little cubes as we scale its
length by a factor .**

What about multiplying something other than length? It follows from
our discussion that if we multiply the ** volume** (rather than all
lengths) of an object with a factor , then we
multiply each linear dimension (length) with and each area with
. If we were to
multiply the ** area** with a factor we'd have to multiply each
linear dimension with and the volume with the cube of
that factor, i.e.,
.
These facts are summarized in the following Table

To use the table find the column that contains a for the
quantity (length, area, or volume) that gets multiplied with a known
factor. The other two quantities are multiplied with the factor given
in that column. So for example, if you know that the area is
multiplied with then you look at the second column headed **factor** and use . The other entries in the column tell you
that the weight is multiplied with
and the length is multiplied with . Remember that quantities like weight go with volume, and
paint goes with area.

Of course, you don't need to memorize that table. Everything flows from the fact that areas get multiplied with the square, and volumes with the cube, of whatever factor multiplies the lengths.