Advanced Algebra
Chapter 1 Answers
Mathematical Models
Part A, Data and Scatter Plots
1. At the National Center for Labor Statistics Web site, look at the labor force participation of persons 16 years old and over for the years 1950 to 1995.
a. Make a scatter plot of the data for males 16 years and older.
b. Draw a trend line for the data. What type of association do you see? [positive association]
c. Predict the number of males that will be employed in the year 2000.
[Approximately 74,000 in thousands, or 74,000,000]d. Make a scatter plot and draw the trend line of the data for the percent of males 16 years and older in the labor force.
e. What type of association do you observe? [negative]
f. Why might the percentage of 16-and-over males in the labor force have decreased from 1950-1995, even though the total male labor force increased?
[Although the number of working males 16 and over increased, the total population of 16-and-over males increased at an even faster rate. Therefore, the ratio 16-and-over working males : 16-and-over males decreased.]
Part B, Patterns and Relationships
2. Look at the pupil / teacher ratios and expenditures per student in public elementary and secondary schools for the years 1955-56 to 1995-96 at the National Center for Educational Statistics Web site.
a. What is the projected pupil/teacher ratio for schools in 1995-96? [17.4]
b. Based on this number, write an equation relating T, the number of teachers in a school, and S, the number of students in a school. [S = 17.4T or T = S : 17.4]
c. Identify the independent variable and the dependent variable. [T is dependent upon S.]
d. Make a table of values for your equation. Include at least six entries.
[Answers will vary; however, they should fit the equation above.]e. Use your equation to predict the number of teachers in a school with 2,001 students. [115]
f. Use your equation to determine how many students you would expect to be enrolled in a school with 50 teachers. [870]
Part C, Functions
1. Look again at the National Center for Educational Statistics Web site to find the percent of 25- to 29-year olds completing high school and college for the years 1940 to 1995.
a. Make a scatter plot of the data for the percent of 25 29-year-olds completing less than 4 years of high school.
b. What type of association do you observe?
[negative association]c. Is the association you saw in b a function? Explain how you know.
[It is a function because exactly one percentage is associated with each year.]
Each day several thousand fireballs--very bright meteors--streak through the earth's atmosphere. A few strike the earth's surface. Because observations and landings of fireballs occur at random locations, they can be analyzed using probabilities.
Part A, Theoretical and Experimental Probability
1. The American Meteor Society keeps track of meteor sightings. At any given time, how likely is it that a sighting will occur?
a. What is the theoretical probability that a fireball observed in March will occur on March 29? What is the experimental probability that a March 1998 fireball reported to the American Meteor Society was observed on March 29?
[1 : 31 ; 4 : 20 = 1 : 5]b. What is the theoretical probability that a fireball will occur between the hours of 6 p.m. (1800) and 6 a.m. (0600) local time? What is the experimental probability that a March 1998 fireball reported to the American Meteor Society occurred between the hours of 6 p.m. and 6 a.m. local time? [1 : 2 ; 1]
c. Explain the discrepancy in question b.
[Answers will vary but should mention that, although fireballs occur at random times, all except rare and extremely bright fireballs are observable only during nighttime hours.]d. Refer to "Frequently Asked Questions" (FAQs) about fireballs. If you find a meteorite, what is the probability that it will belong to the iron class? (See FAQ 13.) [28% or 7 : 25]
e. What is the probability that a meteorite fall (landing) will occur over an ocean or uninhabited land area? (See FAQ 10.) [11 : 12]
f. Use the data in FAQ 10 to estimate the surface area of the earth in square miles. (Use the average of the given numbers of daily "meteorite dropping events.") Explain how you got your answer. [About 219,000,000 mi2. There are about (10 + 50 ) : 2 = 30 events per day over the entire earth, or 30 x 365 x 20,000 = 219,000,000 events in 20,000 years.
1 mi2 : area of earth = 1 event :
219,000,000 events
Therefore, area of earth = 219,000,000 mi2
(The actual area is about 197,000,000 mi2.)]
Part B, Simulation
2. Meteors come from comets and asteroids.
a. What percentage of all Meteors have their origins in asteroids? (See FAQ 7.) [about 5%]
b. Describe how you can take digits in a list of random numbers in pairs to represent meteors with their origins in asteroids and meteors with their origins in comets. Use these random digits to illustrate your method: 7, 3, 3, 4, 0, 9, 1, 9, 0, 2.
[Digits in pairs can represent any of the 100 numbers from 00 to 99. Since meteors of asteroid origin occur 5% of the time, let 5% of the 100 numbers (say, 00, 01, 02, 03, and 04) represent asteroid meteors, and let the other 95% of the 100 numbers (05 to 99) represent comet meteors. Using this method, the given random number 7334091902 represents
that is:
73
34
09
19
02
comet comet comet comet asteroid c. Suppose 10 meteorites fall to earth each day. (The actual number ranges from about 10 to about 50 daily.) Use the following random digits to simulate meteorite falls over 10 days. What is the probability that two or more meteorites with their origins in asteroids will fall to earth in a day?
[Answers may vary. Using the method described in b, the probability is 1 : 10.]
55031 98877 01475 66477 01013 27614 66513 70555 26304 64800 50863 36410 21830 07694 69346 44580 23183 41827 88575 30166 84622 39089 36083 45246 42488 78077 69882 61657 34136 79180 97526 04098 32906 07408 11977 09013 23982 25835 14367 24010
Part C, Making Connections
3. A meteorite struck the earth somewhere in Colorado.
a. What is the probability that it landed in Montrose County? To obtain the information you'll need, open the U.S. map Web site. Scroll to the bottom and click on Colorado.
Hint: To find data on Colorado, look at the top of the page and click on the words State of Colorado. Open the Summary Report and select "Land Area." Then click on "Get the above Selected report." The land area data will appear an the bottom of the screen. Record the data.
To find Montrose County data, go back to the Colorado Counties map and click on "Montrose County." Repeat the steps above to find the land area.
[The probability is area of Montrose County : area of Colorado = 2240.7 mi2 : 103728.8 mi2 = 2.2%.]
Credit card interest rates can be confusing. How can you find the best financial deals as you begin paying for your own purchases? Is credit a good way to pay for goods and services? Let's investigate using the Internet and mathematics!
Part A, Matrices and Data Handling
1. Different credit cards charge different rates of interest and annual fees. To help you decide which factors matter the most, look at the Credit card page in the Financial Calculator site.
a. Set the annual fee = 0, and use the default amount for the rest of the values. These should be set to:
Amount now owed: $5000 Future monthly charges: $275 Future monthly payments: $350 Annual rate: 17% Hit the calculate button, then scroll down to "Schedule of payments, etc." Write the first six months as a matrix and label it Matrix A.
b. What is the dimension of Matrix A? [6 x 5 or 6 by 5]
c. Change the interest rate to 6%. Now look at "Schedule of payments, etc." Write the data for the first 6 months as a matrix and label it Matrix B.
d. What is the dimension of Matrix B? [6 x 5 or 6 by 5]
e. Find Matrix A B. Explain what each column represents.
[The first column represents the change in the balance owed when the interest rate drops from 17% to 6%. The second column, all zeros, shows that there is no change in the monthly purchases. The third column, all zeros, indicates that the monthly payment is the same. The fourth column shows the difference between the interest and fees when the interest rate drops from 17% to 6%. The last column shows the cumulative interest and fees dropping to a final savings of $278 over the 6-month period.]f. What is the dimension of Matrix A B?
[6 x 5 or 6 by 5]g. Under the given circumstances, how much money do you save by paying an interest rate of 6% versus an interest rate of 17%? [$278]
h. Explain how you got your answer to a.
[Students should locate their answer in the cell in the lower right corner.]i. Notice that the dimensions of the matrices A, B, and A - B are all the same. Must this always be the case to add or subtract matrices? Explain.
[Yes, because of the way matrix addition and subtraction are defined.]
Part B, Multiplying Matrices
2. Suppose you make the following charges on your three credit cards over a two-month period:
Card 1 Card 2 Card 3 Month 1 $10 $32 $17 Month 2 $42 $8 $12 a. Write a matrix for the table.
b. What are the dimensions of the matrix? [2 x 3]
c. For each card, you have to pay the amount of your purchase as well as any interest that has accrued. Suppose your monthly interest rates are as follows:
Principle Interest Card 1 1 .01 Card 2 1 .015 Card 3 1 .005 d. Write a matrix for the table.
e. What do the numbers in the first column indicate?
[They indicate that 100% of the amount of purchases will accrue interest.]f. What do the numbers in the second column represent? [The rate of monthly interest]
g. What are the dimensions of the matrix? [3 x 2]
h. Find the matrix product.
i. What do the numbers in the rows and columns of the matrix represent?
[The numbers in the first column represent the principle you owe each month, a total of $121. The numbers in the second column represent the interest you pay each month, a total of $2.03.]
Part C, Making Connections
3. Notice that the dimensions of the matrices in parts a and d
are different. Do the dimensions of two matrices have to be different in
order to multiply them together? Either explain why they do, or give an
example of two matrices that have the same dimensions but that can be multiplied.
[The number of rows of the matrix on the left must equal the number of
columns of the matrix on the right. Any two square matrices can be multiplied,
such as two 3 x 3 matrices.]