3.7 Projects for Chapter 3
Project
Wien's Law
A hot object such as a light bulb or a star radiates energy over a range of wavelengths, but the wavelength with maximum energy is inversely proportional to the temperature of the object. If temperature is measured in
kelvins, and wavelength in micrometers, the constant of proportionality is
2898
. (One micrometer is one
thousandth of a millimeter, or
1
μ
m
=
10
−
6
meter.)
Write a formula for the wavelength of maximum energy,
λ
max
, as a function of temperature,
T
. This formula, called Wien's law, was discovered in 1894. Our sun's temperature is about
5765
K. At what wavelength is most of its energy radiated? The color of light depends on its wavelength, as shown in the table. Can you explain why the sun does not appear to be green? Use Wien's law to describe how the color of a star depends on its temperature. Astronomers cannot measure the temperature of a star directly, but they can determine the color or wavelength of its light. Write a formula for
T
as a function of
λ
max
. Estimate the temperatures of the following stars, given the approximate value of
λ
max
for each. Sketch a graph of
T
as a function of
λ
max
and locate each star on the graph. Project
Halley's Comet
Halley's comet which orbits the sun every
76
years, was first observed in 240 B.C. Its orbit is highly elliptical, so that its closest approach to the Sun (perihelion ) is only
0.587
AU, while at its greatest distance (aphelion ) the comet is
34.39
AU from the Sun. (An AU, or astronomical unit, is the distance from the Earth to the Sun,
1.5
×
10
8
kilometers.)
Calculate the distances in meters from the Sun to Halley's comet at perihelion and aphelion. Halley's comet has a volume of
700
cubic kilometers, and its density is about
0.1
gram per cubic centimeter. Calculate the mass of the comet in kilograms. The gravitational force (in newtons) exerted by the Sun on its satellites is inversely proportional to the square of the distance to the satellite in meters. The constant of variation is
G
m
1
m
2
, where
m
1
=
1.99
×
10
30
kilograms is the mass of the Sun,
m
2
is the mass of the satellite, and
G
=
6.67
×
10
−
11
is the gravitational constant. Write a formula for the force,
F
, exerted by the sun on Halley's comet at a distance of
d
meters. Calculate the force exerted by the sun on Halley's comet at perihelion and at aphelion. Project
World Records
Are world record times for track events proportional to the length of the race? The table gives the men's and women's world records in 2005 for races from 1 kilometer to 100 kilometers in length.
On separate graphs, plot the men's and women's times against distance. Does time appear to be proportional to distance? Use slopes to decide whether the graphs of time versus distance are in fact linear. Both sets of data can be modeled by power functions of the form
t
=
k
x
b
, where
b
is called the fatigue index . Graph the function
M
(
x
)
=
2.21
x
1.086
over the men's data points, and
W
(
x
)
=
2.46
x
1.099
over the women's data. Describe how the graphs of the two functions differ. Explain why
b
is called the fatigue index. Project
Naismith's Number
Fell running is a popular sport in the hills, or fells, of the British Isles. Fell running records depend on the altitude gain over the course of the race as well as its length. The equivalent horizontal distance for a race of length
x
kilometers with an ascent of
y
kilometers is given by
x
+
N
y
, where
N
is Naismith's number (see Naismith's rule ). The record times for women's races are approximated in minutes by
t
=
2.43
(
x
+
9.5
y
)
1.15
, and men's times by
t
=
2.18
(
x
+
8.0
y
)
1.14
. (Source: Scarf, 1998)
Whose times show a greater fatigue index, men or women? (See World Records .) Whose times are more strongly affected by ascents? Predict the winning times for both men and women in a
56
-kilometer race with an ascent of
2750
meters. Project
Elasticity
Elasticity is the property of an object that causes it to regain its original shape after being compressed or deformed. One measure of elasticity considers how high the object bounces when dropped onto a hard surface,
e
=
height bounced
height dropped
(Source: Davis, Kimmet, and Autry, 1986)
The table gives the value of
e
for various types of balls. Calculate the bounce height for each ball when it is dropped from a height of
6
feet onto a wooden floor. Write a formula for
e
in terms of
H
, the bounce height, for the data in part (a). Graph the function from part (b). If Ball A has twice the elasticity of Ball B, how much higher will Ball A bounce than Ball B? Project
Mersenne's Laws
The tone produced by a vibrating string depends on the frequency of the vibration. The frequency in turn depends on the length of the string, its weight, and its tension. In 1636, Marin Mersenne quantified these relationships as follows. The frequency,
f
, of the vibration is
inversely proportional to the string's length,
L
, directly proportional to the square root of the string’s tension,
T
, and inversely proportional to the square root of the string's weight per unit length,
w
. (Source: Berg and Stork, 1982) Write a formula for
f
that summarizes Mersenne's laws. Sketch a graph of
f
as a function of
L
, assuming that
T
and
w
are constant. (You do not have enough information to put scales on the axes, but you can show the shape of the graph.) On a piano, the frequency of the highest note is about
4200
hertz. This frequency is
150
times the frequency of the lowest note, at about
28
hertz. Ideally, only the lengths of the strings should change, so that all the notes have the same tonal quality. If the string for the highest note is
5
centimeters long, how long should the string for the lowest note be? Sketch a graph of
f
as a function of
T
, assuming that
L
and
w
are constant Sketch a graph of
f
as a function of
w
, assuming that
L
and
T
are constant. The tension of all the strings in a piano should be about the same to avoid warping the frame. Suggest another way to produce a lower note. The longest string on the piano in part (c) is
133.5
cm long. How much heavier (per unit length) is the longest string than the shortest string? Project
Damuth's Formula
In 1981, John Damuth collected data on the average body mass,
m
, and the average population density,
D
, for 307 species of herbivores. He found that, very roughly,
D
=
k
m
−
0.75
(Source: Burton, 1998)
Explain why you might expect an animal's rate of food consumption to be proportional to its metabolic rate. (See Example in Rational Exponents for an explanation of metabolic rate.) Explain why you might expect the population density of a species to be inversely proportional to the rate of food consumption of an individual animal. Use Kleiber's rule and your answers to parts (a) and (b) to explain why Damuth's proposed formula for population density is reasonable. Sketch a graph of the function
D
. You do not have enough information to put scales on the axes, but you can show the shape of the graph.
k
=
1
. Project
Self-thinning Law
Studies on pine plantations in the 1930s showed that as the trees grow and compete for space, some of the die, so that the density of trees per unit area decreases. The average mass of an individual tree is a power function of the density,
d
, of the trees per unit area, given by
M
(
d
)
=
k
d
−
1.5
This formula is known as the
−
3
2
self-thinning law . (Source: Chapman and Reiss, 1992)
To simplify the calculations, suppose that a pine tree is shaped like a tall circular cone and that as it grows, its height is always a constant multiple of its base radius,
r
. Explain why the base radius of the tree is proportional to the square root of the area the tree covers. Write
r
as a power function of
d
. Write a formula for the volume of the tree in terms of its base radius,
r
. Use part (b) to write the volume as a power function of
d
. The mass (or weight) of a pine tree is roughly proportional to its volume, and the area taken up by a single tree is inversely proportional to the plant density,
d
. Use these facts to justify the self-thinning law. Sketch a graph of the function
M
. You do not have enough information to put scales on the axes, but you can show the shape of the graph.
k
=
1
. Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later .
