Homework 1
92.450 / 550 Math Modeling
Solutions
1. Given a body with volume v and surface area s, dene = s/v 2/3 .
(a) Find for a sphere and a cube.
4r2
)2/3
r3
3
For the sphere of radius r, = ( 4
For the cube with side length s, =
(
)1/3
4
= ( 4 )2/3 = 3
Homework 4
92.450 / 550 Math Modeling
Solutions
1. (a) In Newtons time the radius of the Earth and the acceleration due to gravity on the Earths surface were
known. Use this information to nd k for the Earth. Use MKS units.
From class, g = k/R2 where R 6.
Homework 6
92.450 / 550 Math Modeling
Due Mar 25
1. The hyperbolic functions sinh and cosh are dened by
sinh x =
1( x
e
2
)
)
1(
ex and cosh x = ex + ex .
2
(a) Sketch the graphs of sinh and cosh on the same axes.
(b) Show sinh = cosh, cosh = sinh and co
Homework 6
92.450 / 550 Math Modeling
Solutions
1. The hyperbolic functions sinh and cosh are dened by
)
)
1(
1(
sinh x = ex ex and cosh x = ex + ex .
2
2
(a) Sketch the graphs of sinh and cosh on the same axes.
(b) Show sinh = cosh, cosh = sinh and cosh2
Homework 7
1. Find
92.450 / 550 Math Modeling
L
n
x2 cos x
L
0
Solutions
dx where n is a positive integer.
Integration by parts twice gives
L
n
x2 cos x
L
0
dx =
2(1)n L3
.
n2 2
2. A man of mass m jumps from a bridge attached to a bungee cord of stiffne
Homework 2
92.450 / 550 Math Modeling
Due Feb 11
1. (a) Using the substitutions from class, show that the initial value problem
A
dh
= R a 2gh ,
dt
h(0) = H
can be written in the nondimensional form
d
= Q ,
d
(0) = 1
and determine Q in terms of the variab
Homework 4
92.450 / 550 Math Modeling
Due March 4
1. (a) In Newtons time the radius of the Earth and the acceleration due to gravity on the Earths surface were
known. Use this information to nd k for the Earth. Use MKS units.
(b) Find the orbital radius o
Homework 7
1. Find
92.450 / 550 Math Modeling
L
n
x2 cos x
L
0
Due April 1
dx where n is a positive integer.
2. A man of mass m jumps from a bridge attached to a bungee cord of stiffness k (units m/t2 ) and length L. Under
the inuence of gravity g he fal
Homework 5
92.450 / 550 Math Modeling
Due Mar 11
1. Find the period of small oscillations of a uniform stick pendulum of length L and mass m with a ball of mass
M at the end.
Hint. The energy of the system is the sum of the energies of the simple and stic
Homework 1
92.450 / 550 Math Modeling
Due Feb 4
1. Given a body with volume v and surface area s, dene = s/v 2/3 .
(a) Find for a sphere and a cube.
(b) Find two differently shaped bodies with the same .
(c) Show that there are bodies which t inside the u
Homework 2
92.450 / 550 Math Modeling
Solutions
1. (a) Using the substitutions from class, show that the initial value problem
dh
A = R a 2gh ,
h(0) = H
dt
can be written in the nondimensional form
d
= Q ,
d
(0) = 1
and determine Q in terms of the paramet
Homework 3
92.450 / 550 Math Modeling
Solutions
1. Harvesting. Let N (t) denote the number of sh at time t which evolves according to the Verhulst model in the
absence of harvesting. If shing reduces N at the constant rate E, then N satises the differenti
Homework 5
92.450 / 550 Math Modeling
Solutions
1. Find the period of small oscillations of a uniform stick pendulum of length L and mass m with a ball of mass
M at the end. Hint. The energy of the system is the sum of the energies of the simple and stick
Homework 8
92.450 / 550 Math Modeling
Solutions
1. SIR Model. Consider the spread of an infectious disease through an initially healthy population N . Let I be the
number of infectives (sick and can transmit the disease) and S the number of susceptibles (