X and the City: Modeling Aspects of Urban Life (12 page)

But the weight
W
of anything is (for a given mean density) proportional to its volume, and its volume is proportional to the cube of its size, that is,
W
∝
L
³
, so
L
²
∝
W
^2/3
, from which we infer that
t
_c
∝
W
^2/3
. And that is our basic result:
the time necessary to adequately cook our turkey is proportional to the two-thirds power of its weight
. We have in fact made use of a very powerful technique in applied mathematics in general (and mathematical modeling in particular):
dimensional analysis
. As seen above, this involves scaling quantities by characteristic units of a system, and in so doing to reveal some fundamental properties of that system.

Where can we go from here? One possible option is to determine the unknown constant of proportionality
a
/
κ
; in principle
κ
can be found (but probably not in any cookbook you possess), but of course
a
is defined in terms of
t
_c
, which doesn’t help us! However, if we have a “standard turkey” of weight
W
₀
and known cooking time
t
₀
, then for any other turkey of weight
W
, a simple proportion gives us

From this the cooking time can be calculated directly. Note that
t
_c
is not a linear function of weight; in fact the cooking time
per pound
of turkey decreases as the inverse cube root of weight, since

Hence doubling the cooking time
t
₀
for a turkey of weight 2
W
₀
may result in an overcooked bird; the result (4.4) implies that
t
_c
= 2
^2/3
t
₀
≈ 1.6
t
₀
should suffice. However, always check the bird to be on the safe side. Note that the units used here, pounds, are really pounds-force, a unit of weight. Generally, pounds proper are units of mass, not weight, but in common usage the word is used to mean weight.

Of course, this is a rather ill-posed question. Are we referring to large tomatoes, which we slice nd put in salads, or those little ones that we find in salad bars? Of course, we can find both sizes, and those in between, in any supermarket or produce store. And I suspect that more tomatoes are consumed during the summer than the rest of the year, for obvious reasons. (Note that we are not including canned tomatoes used in pasta sauce.) Now big tomatoes and little tomatoes share a very import characteristic: they are both tomatoes! I’m going to work with a typical timescale of one week; that is, I might use one large tomato per week in my sandwiches, or consume more of the smaller tomatoes in the same period of time. Therefore I will take the average in the following sense. A fairly big tomato 3 inches in diameter is about 30 times as large by volume as one that is one inch in diameter, so we can use the Goldilocks principle referred to earlier—
is it too large, too small, or just right?
To implement this, we merely take the geometric mean of the volumes. (A reminder: the geometric mean of two positive numbers is the most appropriate measure of “average” to take when the numbers differ by orders of magnitude.) The geometric mean of 1 and 30 is
about 5, so accounting for the range of sizes, we’ll work with 5 “generic” tomatoes eaten per week in the calculations below. Let’s suppose that about a third of the U.S. population of 300 million eat tomatoes regularly, at least during the summer. (I am therefore ignoring those adults and children who do not eat them by dividing the population into three roughly equal groups, again using the Goldilocks principle: fewer than 100% but more than 10% of the population eat tomatoes; the geometric mean is
, or about 1/3.)

Therefore, if on average one third of the population eat 5 tomatoes per week, then halving this to reflect a smaller consumption (probably) during the winter months (except in Florida
), the approximate number of tomatoes eaten every year is

that is, three billion tomatoes. We multiply this by about 2/3 (to account for the proportion of city-dwelling population in the U.S.) to get roughly two billion (with no offense to non-city dwellers, I trust).

I love apples, don’t you? Sometimes though, they contain “visitors.” Suppose that there is a “bug” of some kind in a large spherical apple of radius, say, two inches. We will assume that it is equally likely to go anywhere within the apple (we shall ignore the core). What is the probability that it will be found within a typical “bite-depth” of the surface? Based on my lunchtime observations, I shall take this as ¾ inch, but as always, feel free to make your own assumptions.

The probability
P
of finding the bug within one bite-depth of the surface is therefore the following ratio of volumes (recall that the volume of a sphere is 4
π
/3 times the cube of its radius):

or about 76%.
Ewwwhhh
!

When I was growing up, I loved to visit my grandfather. Despite living in a city (or at least, a very large town) he was able to cultivate and maintain quite a large garden, containing many beautiful plants and flowers. In fact, whenever I asked what any particular flower was called, his reply was always the same: “
Ericaceliapopolifolium!
” I never did find out whether he was merely humoring me, or whether he didn’t know! In addition to his garden, he rented a smaller strip of land (an “allotment”) farther up the road where, along with several others, he grew potatoes, carrots, beans, and other vegetables. I’m
ashamed to admit that I didn’t inherit his love for the art of gardening, much to my parents’ disappointment. It skipped a generation, though; my son has a lovely garden and my son-in-law has a very “green thumb” (of course, neither I nor my grandfather can be held responsible for the latter).

As with the previous chapter, there are various and sundry topics in this one, connected (somewhat tenuously, to be sure) by virtue of being found in a garden or greenhouse. Let’s get specific. Plants grow. My daughter and her family live in northern Virginia, and at the bottom of their garden they have quite a lot of bamboo plants. These can reach great heights, so my question is,

X
=
h
′(
t
): Question:
How fast does bamboo grow?

The growth rate for some types of bamboo plant may be as much as 4 meters (about 13 ft) per day. Let’s work with a more sedate type of bamboo, growing only (!) at the rate of 3 ft/day (about 1 m / day). Just for fun, let’s convert this rate to (i) miles/second, (ii) mph, and (iii) km/decade (ignoring leap years!).