Advanced Algebra
Chapter 7 Answers
Investigating Roots and Powers
To the casual observer, starlight is just an intriguing twinkle in the night sky. To astronomers, starlight is an endless source of information about our universe.
Part A, Inverse, Combined, and Joint Variation;
Part B, Positive Integer Exponents
1. The brightness of a distant light, as you see it, depends on the actual intensity of the light (Is it a tiny 25-watt bulb or a giant spotlight?) as well as its distance from you. The same is true of starlight. A star's brightness, as you see it, depends on the star's absolute brightness and its distance. Absolute brightness is different from a star's magnitude, which is a measure of the intensity of the star's light as you see it.
a. Find the magnitude and distance of the two stars Barnard's Star and Ross 154. [Barnard's: mag 9.5, dist 5.96 l.y.; Ross 154: mag 10.6, dist 9.4 l.y.]
b. Read about magnitude beneath the star table. What does the difference in the magnitudes of the two stars tell you about how much brighter Barnard's Star appears in the night sky than Ross 154 appears? [Barnard's Star appears about 2.5 times brighter than Ross 154.]
c. The absolute brightnesses of Barnard's Star and Ross 154 are almost identical. The difference in their magnitudes is due to the fact that their distances from Earth are different. Brightness varies inversely as the square of an object's distance from the observer. Use this fact and your answer to 1b to find how many times farther from Earth is Ross 154 than is Barnard's Star. Explain your method. [About 1.58 times farther;
]
d. Compare your answer to 1c with the data on star distances you found in the table. Are the data compatible with your answer? [Table: (Ross 154 distance / Barnard's distance) = (9.4 / 5.96) ≈ 1.58; the data give almost identical results and are therefore compatible.]
Part C, Properties of Exponents and Powers;
Part D, Making Connections
2. Moonlight captured on a photograph can be used to estimate the height of a moon crater.
a. The moon's mass is 7.35 X 1022 kg. Its density is 3.34 X 103 kg/m3. Use the fact that density = (mass / volume) to find the moon's volume. [2.2 X 1019 m3]
b. The volume V of a sphere with radius r is given by V = (4 / 3) πr3. Use your answer to a to find the moon's radius in meters. [1.74 X 106 m]
c. Find the moon photo titled L2-7 days (1st quarter. Click on the photo for a detailed image. Locate the crater with a wide shadow on the Web site photo. Use the picture shown here to help locate the crater. With a millimeter ruler, take these measurements from the Web site photo:
s = width of shadow in crater
d = distance of shadow from moon's shadow line
r = radius of moon (you may have to scroll to get this)
[Measurements will vary.]
d. If h represents the height of the crater wall, why does (r / d) = (s / h)?
[The right triangles are similar by AA, so corresponding parts are proportional.]
e. Use the proportion to find h in millimeters. [Answers will vary.]
f. In the Web site photo, what fraction of the radius of the moon r is represented by the height of the crater wall h? [Answers will vary; approximately (1 / 400) = 0.0025]
g. Dimensions on the photo are proportional to actual dimensions on the moon. Use your answers to 2b and 2f to approximate the height of the crater wall. [Answers will vary; 4400 m]
The speed of sound depends on the properties of the medium in which the sound travels.
Part A, Rational Exponents;
Part B, Real-Number Exponents and Modeling
1. The density of a medium is its mass per unit volume. The bulk modulus of a medium is a measure of how the medium reacts to changes in pressure and volume. You can use the density and bulk modulus of a medium to find the velocity of sound in the medium.
a. Click on at least five different elements in the Periodic Table of the Elements. Record the bulk modulus, the density, and the velocity of sound in each element. (You may need to try more than five elements, since bulk modulus is not recorded for all elements in the table.) The multiplier "X 109" has been omitted from the bulk moduli in the table. You should append it to your values. [Answers will vary; Example: aluminum, bulk modulus 75.5 X 109, density 2700 kg/m3, velocity of sound 5100 m/sec]
b. For each element you have chosen, calculate n = (bulk modulus / density) . [Answers will vary; Example: aluminum 27,963,000]
c. Use a power regression to find the equation v = AnB, where v is the velocity of sound in a medium, and n = (bulk modulus / density) for the medium. [Answers will vary but should approximate v = n0.5 (A = 1 and B = 0.5).]
d. The bulk modulus of air is approximately 142,000 (do not append X 109). The density of air is 1.3 kg/m3. Use your power regression to calculate the velocity of sound in air. [Answers will vary depending on student's power regression; 330.5 m/sec]
Part C, Solving Radical Equations;
Part D, Other Radical Equations
2a. Find the density of the element rhodium (Rh) and the speed of sound in rhodium. [density 12,450 kg/m3; velocity of sound 4700 m/sec]
b. The table does not give the bulk modulus (B) of rhodium. Approximate B using the formula v =
(v = velocity of sound, d = density) and solving the resulting radical equation. Write the bulk modulus in the form "B X 109." [275 X 109]
Part E, Making Connections
3. A spectrogram is a graph of a sound wave. Time is graphed on one axis and the frequency of the sound wave (the speed of vibration) on the other.
a. Find the spectrogram of the "clicks" of a sperm whale. Each unit on the horizontal axis represents 1000 Hertz (Hz) (vibrations per second). The vertical axis represents time, with t = 0 at the bottom. Notice that the sound consists of 5 simultaneous clicks, which do not change in frequency as time passes. Estimate the frequency of the click with the lowest frequency. [about 2700 Hz]
b. Shown below is the left portion of a piano keyboard. You can use the keyboard to find the musical scale note that the lowest of the five whale clicks makes.
The frequency of the lowest note on a piano, A1, is 27-1/2 Hz. The frequency of each succeeding note A is found by doubling that of the A to its left. Find the two A's between which the lowest whale click occurs, and give their frequencies. [A7 (1760 Hz) and A8 (3520 Hz)]
c. Moving right, the frequency of each note, black or white, is found by multiplying the frequency of the previous note by
. Study the pattern of black and white notes. Then identify the note that the lowest of the five whale clicks makes. [About halfway between E7 (2637 Hz) and F7 (2793 Hz).]
d. The length of the lowest-click sound wave is 1 ft 9 in. Use v = fl to find the velocity v of sound in water (f = frequency, l = wavelength). [4725 ft/sec]
Some factors such as temperature and humidity affect the weather daily. Others factors affect the weather rarely (e.g., volcanic eruptions) or intermittently, like the phenomenon you'll investigate now.
Part A, Operations with Functions;
Part B, Composite Functions
1. Some meteorologists believe that a disturbance known as "El Niño" may have upset normal weather patterns in 1998.
a. Find and record the median monthly precipitation (1961-1990) for your city or for a city near you for each month January through December. [Data will vary depending on student's location.]
b. Graph the median monthly precipitation figures. Show months on the horizontal axis and precipitation amounts on the vertical axis. Draw a curve connecting the points. [For each month, the sum of the "median" and "El Niño y-values should equal the "1998" y-value. Sample graph:
c. Find the monthly precipitation figures for 1998. To do this, choose your local National Weather Service office ("NWS Offices"). Then select "Climate Data" or "Climatological Data," and finally "Monthly Summaries." Graph the figures on the same axis you used to graph the median figures. Label the two curves. [See sample graph shown in 1b.]
d. For each month, plot a third point on the graph such that the sum of the y-value of the point and the y-value of the "median" point equals the y-value of the "1998" point. Draw a curve connecting the points and label it "El Niño." Graphs will vary. [See sample graph shown in 1b.]
e. Let f represent the median precipitation function for your town and let g represent the El Niño function. Describe the 1998 precipitation function h in terms of f and g. [h = f + g]
Part C, Inverse Relations and Functions;
Part D, Making Connections
2a. Meteorologists have collected enormous amounts of data about factors that influence the weather. Find the contour-map database with this data.
b. This Web site draws weather maps for any region and any date in the past century, displaying data relating to 16 different weather factors. Begin by choosing to display
, contour plots, then press
. Under "Data Entry for Contour Plot," choose a date you're interested in, such as your birth date.
c. Under "Select Parameter," find two weather factors which may be related for the date you've chosen. For example, "Daily Percent of Possible Sunshine" might be affected by "Daily Precipitation." Select one of these parameters. Then under "Select Region," select your state. Choose
.
d. Describe the maps and tell what you learned about your two chosen parameters for the date you chose. [Answers will vary. Students should mention that the requested data is shown using contour maps of their state together with a parameter key (e.g., colors, arrows, contour lines). They should explain how the key works. Finally, they should give specific data on the chosen parameters for the day chosen in their state.]
e. How, if at all, do the maps show how one parameter may be a function of the other? [Answers will vary. Students should describe any relationships between parameters that they can observe on the maps. "Daily Snowfall," for example, might be greater on areas of the map where "Daily Minimum Temperature" is low. If they cannot see any relationships, they should give reasons why such relationships might not be discernible.]