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 \(2898\text{.}\) (One micrometer is one thousandth of a millimeter, or \(1 \mu m = 10^{-6}\) meter.)

Write a formula for the wavelength of maximum energy, \(\lambda_{\text{max}}\text{,}\) as a function of temperature, \(T\text{.}\) 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.

Color

Wavelength (\(\mu\)m)

Red

0.64 – 0.74

Orange

0.59 – 0.64

Yellow

0.56 – 0.59

Green

0.50 – 0.56

Blue

0.44 – 0.50

Violet

0.39 – 0.44

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 \(\lambda_{\text{max}}\text{.}\)

Estimate the temperatures of the following stars, given the approximate value of \(\lambda_{\text{max}}\) for each.

Star

\(\lambda_{\text{max}}\)

Temperature

R Cygni

\(1.115\)

Betelgeuse

\(0.966\)

Arcturus

\(0.725\)

Polaris

\(0.414\)

Sirius

\(0.322\)

Rigel

\(0.223\)

Sketch a graph of \(T\) as a function of \(\lambda_{\text{max}}\) and locate each star on the graph.

Project3.4.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, \(1.5\times 10^8\) 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 \(Gm_1 m_2\text{,}\) where \(m_1 = 1.99\times 10^{30}\) kilograms is the mass of the Sun, \(m_2\) is the mass of the satellite, and \(G = 6.67\times 10^{-11}\) is the gravitational constant. Write a formula for the force, \(F\text{,}\) 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.

Project3.5.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.

Distance (km)

Men’s record (min)

Women’s record (min)

\(1\)

\(2.199\)

\(2.483\)

\(1.5\)

\(3.433\)

\(3.841\)

\(2\)

\(4.747\)

\(5.423\)

\(3\)

\(7.345\)

\(8.102\)

\(5\)

\(12.656\)

\(14.468\)

\(10\)

\(26.379\)

\(29.530\)

\(20\)

\(56.927\)

\(65.443\)

\(25\)

\(73.93\)

\(87.098\)

\(30\)

\(89.313\)

\(105.833\)

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 \(t = kx^b\text{,}\) where \(b\) is called the fatigue index. Graph the function \(M(x) = 2.21x^{1.086}\) over the men’s data points, and \(W(x) = 2.46x^{1.099}\) over the women’s data. Describe how the graphs of the two functions differ. Explain why \(b\) is called the fatigue index.

Project3.6.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 \(x + Ny\text{,}\) where \(N\) is Naismith’s number (see Project 1.6). The record times for women’s races are approximated in minutes by \(t = 2.43(x + 9.5y)^{1.15}\text{,}\) and men’s times by \(t = 2.18(x + 8.0y)^{1.14}\text{.}\) (Source: Scarf, 1998)

Whose times show a greater fatigue index, men or women? (See Project 3.5.)

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.

Project3.7.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,

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.

Type of ball

Bounce height

\(e\)

Baseball

\(0.50\)

Basketball

\(0.75\)

Golfball

\(0.60\)

Handball

\(0.80\)

Softball

\(0.55\)

Superball

\(0.90\)

Tennisball

\(0.74\)

Volleyball

\(0.75\)

Write a formula for \(e\) in terms of \(H\text{,}\) 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?

Project3.8.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, \(f\text{,}\) of the vibration is

inversely proportional to the string’s length, \(L\text{,}\)

directly proportional to the square root of the string’s tension, \(T\text{,}\) and

inversely proportional to the square root of the string’s weight per unit length, \(w\text{.}\) (Source: Berg and Stork, 1982)

Write a formula for \(f\) that summarizes Mersenne’s laws.

Sketch a graph of \(f\) as a function of \(L\text{,}\) 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 \(T\text{,}\) assuming that \(L\) and \(w\) are constant

Sketch a graph of \(f\) as a function of \(w\text{,}\) 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.

Hint: Look at a piano’s strings.

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?

Project3.9.Damuth’s Formula.

In 1981, John Damuth collected data on the average body mass, \(m\text{,}\) and the average population density, \(D\text{,}\) for 307 species of herbivores. He found that, very roughly,

\begin{equation*}
D = km^{-0.75}
\end{equation*}

(Source: Burton, 1998)

Explain why you might expect an animal’s rate of food consumption to be proportional to its metabolic rate. (See Example 3.87 in Section 3.4 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 \(D\text{.}\) You do not have enough information to put scales on the axes, but you can show the shape of the graph.

Hint: Graph the function for \(k = 1\text{.}\)

Project3.10.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, \(d\text{,}\) of the trees per unit area, given by

This formula is known as the \(\dfrac{-3}{2}\) 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, \(r\text{.}\) 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 \(d\text{.}\)

Write a formula for the volume of the tree in terms of its base radius, \(r\text{.}\) Use part (b) to write the volume as a power function of \(d\text{.}\)

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, \(d\text{.}\) Use these facts to justify the self-thinning law.

Sketch a graph of the function \(M\text{.}\) You do not have enough information to put scales on the axes, but you can show the shape of the graph.