Figure 3.8. Equidistance contours and “attraction” boundary for an elliptical city.
Suppose that on a reasonably busy day at the post office, λ = 2. What is the chance that there will be six customers ahead of me when I arrive? (Of course, there is also the question of how quickly on average the counter clerks serve the customers, but that is another issue.)
Then
or just over 1 percent. That’s not bad! And that’s just for starters; we will meet the Poisson distribution again in Chapter 9.
X = Pr: CHANCE ENCOUNTER?
Another probability problem. Suppose that Jack and Jill, freshly arrived in the Big City, each have their own agendas about things to do. They decide—sort of—to do their “own thing” for most of the day and then meet sometime between 4:00 p.m. and 6:00 p.m. to have a bite to eat and discuss the day’s activities. Both of them are somewhat disorganized, and they fail to be more specific than that (after all, there are so many books to look at in Barnes and Noble, or Foyle’s Bookshop in London, who knows when Jack will be ready to leave?) Furthermore, neither cell (mobile) phone is charged, so they cannot ask “where the heck are you?”
Suppose that each of them shows up at a random time in the two-hour period. Jack is only willing to wait for 15 minutes, but Jill, being the more patient of the two (and frankly, a nicer person) is prepared to wait for half an hour. What is the probability they will actually meet? The related question of what will happen if they don’t is beyond the scope of this book, but the first place to start looking might be the local hospital.
This is an example of geometric probability. In the 2-hr square shown in Figure 3.9, the diagonal corresponds to Jack and Jill arriving at the same time at any point in the 2-hr interval. The shaded area around the diagonal represents the “tolerance” around that time, so the probability that they meet is the ratio of the shaded area to the area of the square, namely,
or about one third. Not bad. Let’s hope it works out for them.
X = Q(L, M): BUILDING IN THE CITY
To build housing requires land (L) and materials (M), and if the latter consists of everything that is not land (bricks, wood, wiring, etc.), then we can define a function Q(L, M),
Figure 3.9. The shaded area relative to that of the square is the probability that Jack and Jill meet.
where a > 0 and 0 ≤ λ ≤ 1. This form of function is known as a Cobb-Douglas production function, widely used in economics, and named for the mathematician and U.S. senator who developed the concept in the late 1920s.
If the price of a unit of housing is p, the cost of a unit of land per unit area is l, and the cost of unit materials is m (however all these units may be defined), then the builders’ total profit P is given by
If a speculative builder wishes to “manipulate” L and M to maximize profit, then at a stationary point
Exercise: (i) Show, using equation (3.16), that this set of equations takes the algebraic form
(ii) Show also from equation (3.16) that according to this model, such a speculator “gets his just deserts” (i.e., P = 0)!
We shall use equations (3.15) and (3.17) to eliminate Q, L, and M to obtain
where C(α, λ) is a constant. With two additional assumptions this will enable us to derive a valuable result to be used in Chapter 17 (The axiomatic city). If the cost of materials is independent of location, and land costs increase with population density ρ according to a power law, that is, l ∝ ρκ(κ > 0), then
p = Aργ,
where γ = κλ and A is another constant.
Exercise: Derive equation (3.18)
We conclude this section with a numerical application of equation (3.15) in a related context. It is recommended that the reader consult Appendix 5 to brush up on the method of Lagrange multipliers if necessary.
Suppose that
L now being the number of labor “units” and M the number of capital “units” to produce Q units of a product (such as sections of custom-made fencing for a new housing development). If labor costs per unit are l = $50, capital costs per unit are m = $100, and a total of $500,000 has been budgeted, how should this be allocated between labor and capital in order to maximize production, and what is the maximum number of fence sections that can be produced? (I hope the reader appreciates the “nice” numbers chosen here.)
To solve this problem, note that the total cost is given by
The simplified equation will be the constraint used below. The problem to be solved is therefore to maximize Q in equation (3.19) subject to the constraint (3.20). Using the method of Lagrange multipliers we form the function
The critical points are found by setting the respective partial derivatives to zero, thus:
From equations (3.22a) and (3.22b) we eliminate λ to find that M = 3L/2. From (3.22c) it follows that M = 2500, and hence that L = 3750. Finally, from (3.22a), λ = −4(2500)−3/4 (3750)3/4 ≈ −5.4216, so the unique critical point of F is (2500, 3750, −5.4216). The maximum value is Q(L, M) = 16(2500)1/4 (3750)3/4 ≈ 54,200 sections of fencing.
Question: Can we be certain that this is the maximum value of Q?
Exercise: Show that had we not simplified the constraint equation (3.20), the value for λ would have been λ ≈ −0.1084 but the maximum value of Q would remain the same.
X = An: A MORTGAGE IN THE CITY
The home has been built; now it’s time to start paying for it. It’s been said that if you think no one cares whether you’re alive or not, try missing a couple of house payments. Anyway, let’s set up the relevant difference equation and its solution. It is called a difference equation (as opposed to a differential equation) because it describes discrete payments (as opposed to continuous ones).
Suppose that you have borrowed (or currently owe) an amount of money A0, and the annual interest (assumed constant) is 100I % per year (e.g., if I = 0.06, the annual interest is 6%), compounded m times per year. If you pay off an amount b each compounding period (or due date), the governing first-order nonhomogeneous difference equation is readily seen to be
The solution is
Let’s see how to construct this solution using the idea of a fixed point (or equilibrium value). Simply put, a fixed point in this context is one for which An+1 = An. From equation (3.23) such a fixed point (call it L) certainly exists: L = Bm/I. If we now define an = An − L, then it follows from equation (3.24) that
that is
Thus we have reduced the nonhomogeneous difference equation (3.23) to a homogeneous one. Since a1 = λa0, a2 = λa1 = λ2a0, a3 = λa2 = λ3a0, etc., it is clear that equation (3.25) has a solution of the form an = λna0. On reverting to the original variable An and substituting for L and λ the solution (3.24) is recovered immediately.
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 An = 0 and solve for n. Thus
Exercise: Verify this result. And use it to find out when you will have paid off your mortgage!
Chapter 4
EATING IN THE CITY
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!
X = W: WATERMELON WEIGHT
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?
W1 = 0.99W1 + 0.01W1 (water weight plus pith weight) and
W2 = 0.98W2 + 0.02W2, but the pith weight is unchanged; therefore 0.01W1 = 0.02W2, and so
W2 = W1/2;
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 . . .
X = V: HOW MUCH OF THAT FRUIT IS FRUIT?
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 πr2L, 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 4πr3/3. 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 Vm is the volume of the wiener. If a = 3, for example, then Vb = 8Vm. But a cheap hotdog bun is mostly air; about 90% air in fact!
X = tc: TURKEY TIME
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 Tc. We’ll also define tc as the time required to attain this temperature Tc—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/Tc, t′ = t/tc, 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 κtc/L2. Let’s call this constant a. It follows that
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 ∝ L3, so L2 ∝ W2/3, from which we infer that tc ∝ W2/3. 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 tc, which doesn’t help us! However, if we have a “standard turkey” of weight W0 and known cooking time t0, then for any other turkey of weight W, a simple proportion gives us
From this the cooking time can be calculated directly. Note that tc 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 t0 for a turkey of weight 2W0 may result in an overcooked bird; the result (4.4) implies that tc = 22/3t0 ≈ 1.6t0 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.
X = N: HOW MANY TOMATOES ARE CONSUMED
BY CITY-DWELLERS EACH YEAR?
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 ea
t 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).
X and the City: Modeling Aspects of Urban Life Page 4
