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linear algebra

Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

The number of railroad passengers has been increasing in the United States. The table shows railroad ridership from 1988 to 2000.a. Plot the data and find the median-median line.b. Calculate the residuals.c. What is the root mean square error for this model?Explain what it means in this context.d.
The C notes on a piano (C1- C8) are one octave apart. Their relative frequencies double from one C note to the next.a. If the frequency of middle C (or C4) is 261.6 cycles per second, and the frequency of C5 is 523.2 cycles per second, find the frequencies of the other C notes. b. Even though the
In each case below, use the graph and equation of the parent function to write an equation of the transformed image.a.b. c. d.
A rectangle has a perimeter of 155 inches. Its length is 7 inches more than twice its width. a. Write a system of equations using the information given. b. Solve the system and find the rectangle's dimensions.
ℓ1: 2x - 3y = 9 ℓ2: 2x - 3y = - 1 a. Graph ℓ1 and ℓ2. What is the relationship between these two lines? b. Give the coordinates of one point, A, on ℓ1, and two points, P and Q, on ℓ2. c. Describe the transformation that maps ℓ1 onto ℓ2 and A onto P. Write the equation of the image
Solve each equation in Exercise 1 for x?
Rewrite each exponential equation in logarithmic form using the definition of a logarithm. Then solve for x. (Give your answers rounded to four decimal places.) a. 10x = 0.001 b. 5x = 100 c. 35x = 8 d. 0.4x = 5 e. 0.8x = 0.03 f. 17x = 0.5?
Graph each equation. Write a sentence explaining how the graph compares to the graph of either y = 10x or y = log x. a. y = log (x + 2) b. y = 3 log x c. y = - log x - 2 d. y = 10 x + 2 e. y = 3 (10x) f. y = - ( 10x) - 2?
Classify each statement as true or false. If false, change the second part to make it true. a. If 6x = 12, then x = log12 6. b. If log2 5 = x, then 5x = 2. c. If 2 · 3x = 11, then x = log11/2 log3. d. If x = log7/log3, then x = log7 3
The function g(x) = 23(0.94)x gives the temperature in degrees Celsius of a bowl of water x minutes after a large quantity of ice is added. After how many minutes will the water reach 5°C?
Assume the United States's national debt can be estimated with the model y = 0.051517(1.1306727)x, where x represents the number of years since 1900 and y represents the debt in billions of dollars. a. According to the model, when did the debt pass $1 trillion ($1000 billion)? b. According to the
Carbon-14 is an isotope of carbon that is formed when radiation from the sun strikes ordinary carbon dioxide in the atmosphere. Trees, which get their carbon dioxide from the air, contain small amounts of carbon-14. Once a tree is cut down, no more carbon-14 is formed, and the amount that is
Crystal looks at an old radio dial and notices that the numbers are not evenly spaced. She hypothesizes that there is an exponential relationship involved. She tunes the radio to 88.7 FM. After six "clicks" of the tuning knob, she is listening to 92.9 FM. a. Write an exponential model in
Change the form of each expression below using properties of logarithms or exponents, without looking back in the book. Name each property or definition you use. a. gh+k b. log s + log t c. (w/(v d. log h/k e. (js)t f. logbg g. n(km h. logst / logsu i. wt ws j. p-h
Carbon-11 decays at a rate of 3.5% per minute. Assume that 100% is present at time 0 min. a. What percentage remains after 1 min? b. Write the equation that expresses the percentage of carbon-11 remaining as a function of time. c. What is the half-life of carbon-11? d. Explain why carbon-11 is not
Find an equation to fit each set of data. a. x y 1..................8 4.................17 6.................23 7.................26 b. x y 0.....................2 3...................54 4...................162 6.................1458
Describe how each function has been transformed from the parent function y = 2x or y = log x. Then graph the function. a. y = - 4 + 3(2)x - 1 b. y = 2 - log(x/3)?
Answer true or false. If the statement is false, explain why or give a counterexample. a. A grade of 86% is always better than being in the 86th percentile. b. A mean is always greater than a median. c. If the range of a set of data is 28, then the difference between the maximum and the mean must
A driver charges $14 per hour plus $20 for chauffeuring if a client books directly with her. If a client books her through an agency, the agency charges 115% of what the driver charges plus $25.a. Write a function to model the cost of hiring the driver directly. Identify the domain and range.b.
Find the missing values. Then answer the questions to learn about some properties of logarithms. Find the values of 2a-f. a. a = log 18 b. b = log 71 c. c = a + b d. d = antilog c e. e = 18 · 71 f. f = log (18 · 71)? g. Describe any patterns you see. h. Complete the statement "log a + log b = ?
Find the values of 3a-f. a. a = log 58 b. b = log 22 c. c = a - b d. d = antilog c e. e = 58/22 f. ( = log (58/22) g. Describe any patterns you see. h. Complete the statement "log a - log b = ? . i. The property you discovered is the quotient property of logarithms. Explain why this property
Find the value of 4a-d. a. a = log 13 b. b = 3.6a c. c = antilog b d. d = 133.6 e. Compare the values you found. Describe any relationship you see. f. Complete the statement "log ab = ? . g. This property is called the power property of logarithms. How would you write log (a in terms of log a?
Determine whether each equation is true or false. If false, rewrite one side of the equation to make it true. Check your answer on your calculator. a. log 3 + log 7 = log 21 b. log 5 + log 3 = log 8 c. log 16 = 4 log 2 d. log 5 - log 2 = log 2.5 e. log 9 - log 3 = log 6 f. log (7 = log 7/2 g. log
The half-life of carbon-14, which is used in dating archaeological finds, is 5730 yr. a. Assume that 100% of the carbon-14 is present at time 0 yr, or x = 0. Write the equation that expresses the percentage of carbon-14 remaining as a function of time. (This should be the same equation you found in
This table lists the consecutive notes from middle C to the next C note. This scale is called a chromatic scale and it increases in 12 steps, called half-tones. The frequencies measured in cycles per second, or hertz (Hz), associated with the consecutive notes form a geometric sequence, in which
Use the properties of logarithms and exponents to solve these equations. a. 5.1x = 247 b. 17 + 1.25x = 30 c. 27(0.93x) = 12 d. 23 + 45(1.024x) = 147?
The altitude of an airplane is calculated by measuring atmospheric pressure on the surface of the airplane. This pressure is exponentially related to the plane's height above Earth's surface. At ground level, the pressure is 14.7 pounds per square inch (abbreviated lb/in.2, or psi). At an altitude
Prove that these statements of equality are true. Take the logarithm of both sides, then use the properties of logarithms to re-express each side until you have two identical expressions. a. 10n + p = (10n)(10p) b. 10d / 10e = 10d - e?
Quinn starts treating her pool for the season with a shock treatment of 4 gal of chlorine. Every 24 h, 15% of the chlorine evaporates. The next morning, she adds 1 qt (1/4 gal) of chlorine to the pool, and she continues to do so each morning. a. How much chlorine is there in the pool after one day
Find these functions. a. Find an exponential function that passes through the points (4, 18) and (10, 144). b. Find a logarithmic function that passes through the points (18, 4) and (144, 10)?
The Highland Fish Company is starting a new line of frozen fish sticks. It will cost $19,000 to set up the production line and $1.75 per pound to buy and process the fish. Highland Fish will sell the final product at a wholesale cost of $1.92 per pound. a. Write a cost function and an income
Sketch the graph of (4(x + 5) )2 + (y - 8 / 2)2 = 1. Give coordinates of a few points that define the shape?
Solve each equation. Round to the nearest hundredth. a. x5 = 3418 b. (x - 5.1)4 = 256 c. 7.3x6 + 14.4 = 69.4?
Solve each equation. Check your answers by substituting your answer for x. a. 800 = 10x b. 2048 = 2x c. 16 = 0.5x d. 478 = 18.5(10x) e. 155 = 24.0(1.89x) f. 0.0047 = 19.1(0.21x)?
Suppose you invest $3000 at 6.75% annual interest compounded monthly. How long will it take to triple your money?
The length of time that milk (and many other perishable substances) will stay fresh depends on the storage temperature. Suppose that milk will stay fresh for 146 hours in a refrigerator at 4°C. Milk that is left out in the kitchen at 22°C will keep for only 42 hours. Because bacteria grow
The equation ((x) = 12000 / 1 + 499(1.09)-x gives the total sales x days after the release of a new video game. Find each value and give a real-world meaning. a. f (20) b. f (80) c. x when f (x) = 6000 d. Show the steps to solve 5c symbolically? e. Graph the equation in a window large enough for
The intensity of sound, D, measured in decibels (dB) is given by the formulaWhere I is the power of the sound in watts per square centimeter (W/cm2) and 10-16 W/cm2 is the power of sound just below the threshold of hearing. a. Find the number of decibels of a 10 -13 W/cm2 whisper. b. Find the
The table gives the loudness of spoken words, measured at the source, and the maximum distance at which another person can recognize the speech. Find an equation that expresses the maximum distance as a function of loudness. a. Plot the data on your calculator and make a rough sketch on your
A container of juice is left in a room at a temperature of 74°F. After 8 minutes, the temperature is recorded at regular intervals.a. Plot the data using an appropriate window. Make a rough sketch of this graph. b. Find an exponential model for temperature as a function of time. (This curve is
In clear weather, the distance you can see from a window on a plane depends on your height above Earth, as shown in the table at right.a. Graph various combinations of x, y, log x, and log y until you find a combination that linearizes the data.b. Use your results from 9a to find a best-fit
Evaluate each expression without using a calculator. Then check your work with a calculator. a. 4 - 2 b. (-3) - 1 c. (1/5)-3 d. 491/2 e. 64-1/3 f. (9/16)3/2 g. -70 h. (3)(2)2 i. (0.6-2)-1/2
A new incentive plan for the Talk Alot long-distance phone company varies the cost of a call according to the formula cost = a + b log t, where t represents time in minutes. When calling long distance, the cost for the first minute is $0.50. The cost for 15 min is $3.44.a. Find the a-value in the
A "learning curve" describes the rate at which a task can be learned. Suppose the equationpredicts the time t (in number of short daily sessions) it will take to achieve a goal of typing N words per minute (wpm). a. Using this equation, how long should it take someone to learn to type 40 wpm? b. If
All humans start as a single cell. This cell splits into two cells, then each of those two cells splits into two cells, and so on. a. Write a recursive formula for cell division starting with a single cell. b. Write an explicit formula for cell division. c. Sketch a graph to model the formulas in
Rewrite each expression in another form. a. log x + log y b. log z/v c. (7x2.1)(0.3x4.7) d. log wk e. 5(x f. log5 t?
Use the properties of exponents and logarithms to solve each equation. Confirm your answers by substituting them for x. a. 4.7x = 28 b. 4.7x2 = 2209 c. logx 2.9 = 1.25 d. log3.1 x = 47 e. 7x2.4 = 101 f. 9000 = 500(1.065)x g. log x = 3.771 h. 5(x3 = 47?
Solve for x. Round your answers to the nearest thousandth. a. = 2x + 1 b. 4x2.7 = 456 c. 734 =11.2(1.56)x d. f(f -1(x)) = 20.2 e. 147 = 12.1(1 + x)2.3 f. 2(x-3 + 4.5 = 16?
Once a certain medicine is in the bloodstream, its half-life is 16 h. How long (to the nearest 0.1 h) will it be before an initial 45 cm3 of the medicine has been reduced to 8 cm3?
Given f (x) = (4x - 2)1/3 - 1, find: a. f (2.5) b. f -1(x) c. f -1(-1) d. f(f -1(12))?
Find the equation of an exponential curve through the points (1, 5) and (7, 32)?
Draw the inverse of f (x), shown at right?
Your head gets larger as you grow. Most of the growth comes in the first few years of life, and there is very little additional growth after you reach adolescence. The estimated percentage of adult size for your head is given by the formula y = 100 - 80(0.75)x, where x is your age in years and y is
Is (x1/m)n always equivalent to (xn)1/m? Try graphing Y1 = (x1/m)n and Y2 = (xn)1/m for various integer values of m and n. Make sure you try positive and negative values for m and n, as well as different combinations of odd and even numbers. Check to see if the expressions are equal by inspecting
You have learned to use the point-ratio form to find an exponential curve that fits two data points. You could also use the general exponential equation y = abx and your knowledge of solving systems of equations to find an appropriate exponential curve. For example, you can use the general form to
As you become more familiar with a slide rule, you might discover other shortcuts. For example, here is another way to multiply 5 7: Line up the 5 on the top scale with 10 on the bottom scale, then find the number on the top scale that is directly above 7. The answer you find is 3.5, which you then
Use these matrices to do the arithmetic problems 1a-d. If a particular operation is impossible, explain why?a.b. c. d. e. [D] [C] f. [A] [D]
Yolanda, Myriam, and Xavier have a small business producing handmade shawls and blankets. They spin the yarn, dye it, and weave it. A shawl requires 1 h of spinning, 1 h of dyeing, and 1 h of weaving. A blanket needs 2 h of spinning, 1 h of dyeing, and 4 h of weaving. They make a $16 profit per
Graphs 11a-c were produced by a recursive formula in the formu0 = aun= un - 1 (1 + p) + d where n ( 1For each case, tell if a, p, and d are greater than zero (positive), equal to zero, or less than zero (negative).a.b. c.
The graph of y = f (x) is shown at right.a. Find f (-3).b. Find x such that f (x) = 1.c. How can you use the graph to tell whether or not f is a function?d. What is the domain of f?e. What is the range of f?
Last semester, all of Ms. Nolte's students did projects. One-half of the students in her second-period class investigated fractals, one-fourth of the students in that class did research projects, and the remaining students conducted surveys and analyzed their results. In her third-period class,
Canada's oil production has increased over the last half century. This table gives the production of oil per day for various years.a. Define variables and make a scatter plot of the data. b. Find M1, M2, and M3, and write the equation of the median-median line. c. Use the median-median line to
Solve. a. log 35 + log 7 = log x b. log 500 - log 25 = log x c. log (1/8 = x log8 d. 15(9.4)x = 37000 e. f. log6 342 = 2x
Suppose the signal strength in a fiber-optic cable diminishes by 15% every 10 mi. a. What percentage of the original signal strength is left after a stretch of 10 mi? b. Create a table of the percentage of signal strength remaining in 10 mi intervals, and make a graph of the sequence. c. If a phone
The graph of an exponential function passes through the points (4, 50) and (6, 25.92). a. Find the equation of the exponential function. b. What is the rate of change? Does the equation model growth or decay? c. What is the y-intercept? d. What is the long-run value?
This data set gives the weights in kilograms of the crew members participating in the 2002 Boat Race between Oxford University and Cambridge University.(www.cubc.org.uk and www.ourcs.org)a. Make a histogram of these data. b. Are the data skewed left, skewed right, or symmetric? c. Identify any
Consider the graph of y = x2. Let matrix [P], which organizes the coordinates of the points shown, represent five points on the parabola.a. Describe the transformation(s) that would give five corresponding points on the graph of y = (x - 5)2- 2. Sketch the graph. b. Describe the transformation(s)
Find the inverse, if it exists, of each matrix?a.b. c. d.
This table gives the mean population per U.S. household for various years from 1890 to 1998.a. Define variables and find a linear equation that fits these data. b. Write and solve a problem that you could solve by interpolation with your line of fit. c. Write and solve a problem that you could
Solve each system by using row reduction?a.b.
Solve each system by using an inverse matrix?a.b.
Identify each system as consistent and independent (has one solution), inconsistent (has no solution), or consistent and dependent (has infinitely many solutions). a. b. c. d.
Graph the feasible region of each system of inequalities. Find the coordinates of each vertex. Then identify the point that maximizes the given expression.a.Maximize: 1.65x + 5.2y b. Maximize: 6x + 7y
Heather's water heater needs repair. The plumber says it will cost $300 to fix the unit, which currently costs $75 per year to operate. Or Heather could buy a new energy-saving water heater for $500, including installation, and the new heater would save 60% on annual operating costs. How long would
A particular color of paint requires a mix of five parts red, six parts yellow, and two parts black. Thomas does not have the pure colors available, but he finds three pre-mixed colors that he can use. The first is two parts red and four parts yellow; the second is one part red and two parts black;
Interlochen Arts Academy 9th and 10th graders are housed in three dormitories: Picasso, Hemingway, and Mozart. Mozart is an all-female dorm, Hemingway is an all-male dorm, and Picasso is coed. In September, school started with 80 students in Mozart, 60 in Picasso, and 70 in Hemingway. Students are
Russell collected data similar to Karina's at Powder Hill Resort. He found that 86% of the skiers planned to ski the next day and 92% of the snowboarders planned to snowboard the next day.a. Draw a transition diagram for Russell's information.b. Write a transition matrix for the same information.
Lisa Crawford is getting into the moving-truck rental business in three nearby counties. She has the funds to buy about 100 trucks. Her studies show that 20% of the trucks rented in Bay County go to Sage County, and 15% go to Thyme County. The rest start and end in Bay County. From Sage County, 25%
Fly-Right Airways operates routes out of five cities as shown in the route map below. Each segment connecting two cities represents a round-trip flight between them. Matrix [M] displays the information from the map in matrix form with the cities, A, B, C, D, and E, listed in order in the rows and
Solve this system using either substitution or elimination?
Solve the equation 2x + 3y = 12 for y and then graph it?
The table at right shows the number of cellular telephone subscribers in the United States from 1985 to 2000.a. Create a scatter plot of the data. b. Find an exponential function to model the data. c. Use your model to predict the number of subscribers in 2003. Do you think this is a realistic
The equation y = 20 log (x/0.00002) measures the intensity of a sound as a function of the pressure it creates on the eardrum. The intensity, y, is measured in decibels (dB), and the pressure, x, is measured in Pascals (Pa). a. What is the intensity of the sound of a humming refrigerator, if it
Complete this transition diagram:
Matrix [M] represents the vertices of (ABC.a. Name the coordinates of the vertices and draw the triangle. b. What matrix represents the image of (ABC after a translation down 4 units? c. What matrix represents the image of (ABC after a translation right 4 units?
During a recent softball tournament, information about which side players bat from was recorded in a matrix. Row 1 represents girls and row 2 represents boys. Column 1 represents left-handed batters, column 2 represents right-handed batters, and column 3 represents those who can bat with either
A mixture of 40 mL of NO and 200 mL of N2O2 is heated. During each second at this new temperature, 10% of the NO changes to N2O2 and 5% of the N2O2 changes to NO.a. Draw a transition diagram that displays this information.b. Write a transition matrix that represents the same information. List NO
In many countries, more people move into the cities than out of the cities. Suppose that in a certain country, 10% of the rural population moves to the city each year and 1% of the urban population moves out of the city each year. a. Draw a transition diagram that displays this information. b.
Recall the matrix [A] on page 302 that represents the number of math, science, and history textbooks sold at the main and branch campus bookstores this week.a. Explain the meaning of the value of a32. b. Explain the meaning of the value of a21. c. Matrix [B] represents last week's sales. Compare
The three largest categories of motor vehicles are sedan, SUV, and minivan. Suppose that of the buyers in a particular community who now own a minivan, 18% will change to an SUV and 20% will change to a sedan. Of the buyers who now own a sedan, 35% will change to a minivan and 20% will change to an
Look back at the calculations in Example B. Calculate how many students will choose each dessert in the fourth week by multiplying these matrices:
Recall the ice cream and frozen yogurt problem from Example B. Enter these matrices into your calculator, and use them to find the long-run values for the number of students who choose ice cream and the number of those who choose frozen yogurt. Explain why your answer makes sense.
A spider is in a building with three rooms. The spider moves from room to room by choosing a door at random. If the spider starts in room 1 initially, what is the probability that it will be in room 1 again after four room changes? What happens to the probabilities in the long run?
A researcher studies the birth weights of women and their daughters. The weights were split into three categories: low (below 6 lb), average (between 6 and 8 lb), and high (above 8 lb). This transition diagram shows how birth weights changed from mother to daughter.a. Write a transition matrix that
Read the Consumer Connection about Universal Product Codes and answer these questions.a. Write a 12 1 matrix that can be multiplied on the left by a UPC to find the sum of each digit in an even position and 3 times each digit in an odd position.b. Use the matrix from 13a to check the following four
A system of equations that has at least one solution is called consistent. A system of equations that has no solutions is called inconsistent. A system with infinitely many solutions is called dependent. A system of equations that has exactly one solution is called independent. Follow the steps in
For each segment shown in the figure at right, write an equation in point-slope form for the line that contains the segment. Check your equations by graphing them on your calculator.
If logp x = a and logp y = b, find a. logp xy b. logp x3 c. logp y2/x. d. logp2y e. logp(x f. logm xy?
Solve this system of equations for x, y, and z.
Find the missing values.a. [13 23] + [-6 31] = [x y]b.c. d. e.

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