Intro. to Math. Modeling
PROBLEM SET 4
Due February 14, 2005
1. A population is divided into three age groups: 014 years, 1539 years, 40 or older,
as described in class. In that model, we take b1 = b3 = 0, b2 = .4, d1 = .03, d2 = .1, d3 = .3.
Show that
Intro. to Math. Modeling
PROBLEM SET 11
Due April 20, 2005
Note: This is the last graded problem set. A set of review problems will be handed out next week.
1. (a) A mass of ten grams is attached to a spring and allowed to hang without moving in the Earth
Intro. to Math. Modeling PROBLEM SET 6 Due February 28, 2005
1. Consider the linear system
dx
= 2x,
dt
dy
= x 3y.
dt
x
. What is
y
the matrix A? What are its eigenvalues? Is the equilibrium (0, 0) stable or
unstable?
(b) Write the general solution of the
Intro. to Math. Modeling
REVIEW PROBLEMS CONTINUED
April 20, 2005
1. A certain oscillator of mass m has the potential function V (x) = x2 + 2x3 x4.
(a) What is the energy integral of the oscillator?
(b) What is the equation of motion ( mass times accelera
Intro. to Math. Modeling
PROBLEM SET 10
Due April 13, 2005
1. Consider a chemical reaction where A and B combine to produce C with a rate
constant k+ , and simultaneously C can decompose into A and B with a rate constant k .
k+
In symbols
A + B C.
k
(a) T
Case Study 1: The bucket brigade production line
My friend Mike owns a small manufacturing operation in the Garment District of Manhattan.
He called me recently to describe the following problem with his company:
One of the products Mikes company manufact
Intro. to Math. Modeling
PROBLEM SET 7
Due March 9, 2005
1. Consider a twoworker bucket brigade (BB) of the kind we are studying. Worker
1 takes 3 hours to produce a widget, worker 2 takes 1 hour. Assuming the length of the
production line is 1 and time
Intro. to Math. Modeling
PROBLEM SET 5
Due Wednesday, February 23, 2005
1. Problem 49.1, page 227 of text. Explain the interactions by drawing a depressenhance diagram.
2. Consider the hostparasite interaction in which the host growth rate is density de
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Intro. to Math. Modeling
PROBLEM SET 9
Due April 6, 2005
1. Lincoln tunnel data given on page 387 of text for velocity versus density is plotted
in the gure below aned compared with the relation u = a ln(max /) with a = 18 mph
and max = 225 cars per mile.
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Intro. to Math. Modeling
PROBLEM SET 8
Due March 30, 2005
1. The trac ow can be modeled with q = 70(1 /377) vehicles per hour. The
trac is moving at a constant speed and a constant density of 250 vehicles/mile. Because
of an overturned truck, cars start t
Intro. to Math. Modeling
REVIEW PROBLEMS
April 20, 2005
1. A certain species has a birth rate of 250 per 1000 every year, and a death rate of 220 per 100 per
year. The present population is 5000
(a) Let N be the population in 1000s and let time t be measu
Intro. to Math. Modeling
PROBLEM SET 3
Due February 7, 2005
1. Problem2 37.2 of text.
2. Problem 37.5 of text.
3. Prove that the rate of growth or decay of the population in the continuous logistic
model is monotonic, that is d2 N/dt2 > 0 or < 0 for all t
Notes on the bucketbrigade production line
We consider a production line which produces widgets. The line consists of a series of
machines where operations are performed. We take the line to have unit length, and the
position of the k th worker on the li