X and the City: Modeling Aspects of Urban Life (11 page)

Exercise:
Verify the solution (3.24) by direct substitution into (3.23).

Here is a natural question to ask:
How long will it be before we pay off the mortgage?
The answer to this is found by determining the number of pay periods
n
that are left (to the nearest preceding integer; there will generally be a remainder to pay off directly). The time (in years) to pay off the loan is just
n
/
m
(usually
m
= 12 of course). To find it, we set
A
_n
= 0 and solve for
n
. Thus

Exercise:
Verify this result. And use it to find out when you will have paid off
your
mortgage!

The first time I visited a really big city on more than a day trip, I wondered where people found groceries. I had just left home to start my time as an undergraduate in London, but at home I could either ride my bicycle or use my father’s car to drive to the village store or visit my friends (not always in that order). In London I used the “tube”—the underground transport system (the subway) and sometimes the bus, but I walked pretty much everywhere else. I soon found out where to get groceries, and in retrospect realized that there are some quite interesting and amusing mathematics problems associated with food, whether it’s in a city or in the middle of the country. What follows is a short collection of such items connected by a common theme—eating!

A farmer harvested ten tons of watermelons and had them delivered by truck to a town 30 miles away. The trip was a hot and dusty one, and by the time the destination was reached, the watermelons had dried out somewhat; in fact their water content had decreased by one percentage point from 99% water by weight to 98%.

Question:
What was the weight of the watermelons by the time they arrived in the town?

the weight of the watermelons upon arrival is now only five tons. Very surprising, but it shows that a small percentage of a large number can make quite a difference . . .

Suppose that Kate feels like having a healthy snack, and decides to eat a banana. Mathematically, imagine it to be a cylinder in which the length
L
is large compared with the radius
r
. Suppose also that the peel is about 10% of the radius of the original banana. Since the volume of her (now) idealized right circular banalinder (or cylinana) is
πr
²
L
, she loses 19% of the original volume when she peels it (1 − (0.9)
²
= 0.81). Okay, now let’s do the same thing for a spherical orange of volume 4
πr
³
/3. The same arguments with the peel being about 10% of the radius give a 27% reduction in the volume (1 − (0.9)
³
= 0.73). We might draw the conclusion in view of this that it is not very cost-effective to buy bananas and oranges, so she turns her stomach’s attention to a peach. Now we’re going to ignore the thickness of the peach skin (which I eat anyway) in favor of the pit. We’ll assume that it is a sphere, with radius 10% of the peach radius. Then the volume of the pit is 10
⁻³
of the volume of the original peach; a
loss of only 0.1%. What if it were 20% of the peach radius? The corresponding volume loss would be only 0.8 %. These figures are perhaps initially surprising until we carry out these simple calculations [
11
]. But is a banana a fruit or an herb? Inquiring minds want to know.

Meanwhile, the neighbor’s hotdogs are cooking. How much of the overall volume of a hotdog is the meat? Consider a cylindrical wiener of length
L
and radius
r
surrounded by a bun of the same length and radius
R
=
ar
, where
a
> 1. If the bun fits tightly, then its volume is

where
V
_m
is the volume of the wiener. If
a
= 3, for example, then
V
_b
= 8
V
_m
. But a cheap hotdog bun is mostly air; about 90% air in fact!

Question:
How long does it take to cook a turkey (without solving an equation)?

Let’s consider a one-dimensional turkey; these are difficult to find in the grocery store. Furthermore, you may object that a spherical turkey is much more realistic than a “slab” of turkey, and you’d be correct! A spherical turkey might be a considerable improvement. However, the equation describing the diffusion of heat from the exterior of a sphere (the oven) to the interior can be easily converted by a suitable change of variables to the equation of a slab heated at both ends, so we’ll stick with the simpler version.

The governing equation is the so-called heat or diffusion equation (discussed in more detail in
Appendix 10
)

where
T
is the temperature at any distance
x
within the slab at any time
t
;
κ
is the coefficient of thermal diffusivity (assumed constant), which depends on the thermal properties and density of the bird; and
L
is the size (length) of the turkey. This equation, supplemented by information on the temperature of the turkey when it is put in the oven and the oven temperature, can be solved
using standard mathematical tools, but the interesting thing for our purposes is that we can get all the information we need without doing that. In this case, the information is obtained by making the equation above dimensionless. This means that we define new variables for which (i) the dimensions of time and length are “canceled,” so to speak, and (ii) the temperature is defined relative to the interior temperature of the bird when it is fully cooked. We’ll call this temperature
T
_c
. We’ll also define
t
_c
as the time required to attain this temperature
T
_c
—the cooking time. It is this quantity we wish to determine as a function of the size of the bird. The advantage of this formulation is that we don’t have to repeat this calculation for each and every turkey we cook: indeed, as we will see, with a little more sophistication we can express the result in terms that are independent of the size of the turkey.

To proceed with the “nondimensionalization” let
T
′ =
T/T
_c
,
t
′ =
t/t
_c
, and
x
′ =
x/L
. Using the chain rule for the partial derivatives, equation (4.2) in the new variables takes the form

Has anything at all been accomplished? Indeed it has. Since both derivatives are expressed in dimensionless form, then so must be the constant
κt
_c
/L
²
. Let’s call this constant
a
. It follows that