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

For each angle in standard position given, identify the quadrant that the angle's terminal side lies in, and name a coterminal angle. Then convert each angle measure from radians to degrees, or vice versa. a. 60o b. 4π/3 c. 330o d. -π/4
Prove that each of these identities is true. You may use any of the identities that have been proved in this chapter.a. secA - sin A tan A = cos Ab. 1/sin2A - 1/tan2A = 1c.
A mass hanging from a spring is pulled down 2 cm from its resting position and released. It makes 12 complete bounces in 10 s. At what times during the first 3 s was the mass 0.5 cm below its resting position?
These data give the ocean tide heights each hour on November 17, 2002, at Saint John, New Brunswick, Canada.a. Create a scatter plot of the data.b. Write a function to model the data, and graph this function on the scatter plot from 12a.c. What would you estimate the tide height to have been at
Find exact values of the sine and cosine of each angle in Exercise 1. In Exercise 1 a. 60o b. 4π/3 c. 330o d. -π/4
State the period of the graph of each equation, and write one other equation that has the same graph. a. = 2sin (3(x-π/6)) b. y = -3 cos 4x c. y = sec 2x d. y = tan (-2x) + 1
For the sinusoidal equations in 3a and b, state the amplitude, phase shift, vertical translation, and frequency. Then sketch a graph of one complete cycle.
Write an equation for each graph.a.b. c. d.
Identify the domain and range of cos y = x and y = cos-1 x.
Find these values without using your calculator. Then verify your answers with your calculator.a.b. c.
Write an equation for a transformation of y = sin x that has a reflection across the x-axis, amplitude 3, period 8π, and phase shift π/2.
Consider the geometry-software diagram at right. You have seen that in a unit circle, the length AB has the same value as cos Î¸. The lengths AF, GI, AD, AI, and CF correspond to other trigonometric values of Î¸. Decide which segment length equals each of the values sin
You have seen how to find the exact value of cos π/12 by rewriting the expression as cos (π/4, π/6), and using a trigonometric identity to expand and evaluate. How can you find the exact value of sin 11π/12?
You are now familiar with angle measures in degrees and radians, but have you ever heard of gradians? Research the gradian angle measure. Explain how it compares to radians and degrees, when and where it was used, and any advantages it might have. Can you find any other units that measure angle or
The path traced by a fixed point, P, on a moving wheel is called a cycloid and is shown below. A cycloid can be defined with parametric equations. Use the diagram to derive parametric equations for x and y, the coordinates of point P.
In Exercise 13b, you developed the formula cos2 A = 1+cos2A/2. Use this formula to develop the half-angle formula for cosine. Begin by taking the square root of both sides, then substitute θ/2 for A. Use the formula from Exercise 13a for sin2 A and a similar process to develop the half-angle
Find each value. a. Find the sum of the first 1000 positive integers (the numbers 1 through 1000). b. Find the sum of the second 1000 positive integers (1001 through 2000). c. Guess the sum of the third 1000 positive integers (2001 through 3000). d. Now calculate the sum for 10c. e. Describe a way
Suppose y = 65 + 2(x - 1) is an explicit representation of an arithmetic sequence, for integer values x ≥ 1. Express the partial sum of the arithmetic series as a quadratic expression, with x representing the term number.
It takes 5 toothpicks to build the top trapezoid shown at right.You need 9 toothpicks to build 2 adjoined trapezoids and 13 toothpicks for 3 trapezoids.a. If 1000 toothpicks are available, how many trapezoids will be in the last complete row?b. How many complete rows will there be?c. How many
If an object falls from rest, then the distance it falls during the first second is about 4.9 m. In each subsequent second, the object falls 9.8 m farther than in the preceding second.a. Write a recursive formula to describe the distance the object falls during each second of free fall.b. Find an
Consider these two geometric sequences: i. 2, 4, 8, 16, 32, ... ii. 2, 1, 1/2, 1/4, 1/8,... a. What is the long-run value of each sequence? b. What is the common ratio of each sequence? c. What will happen if you try to sum all of the terms of each sequence?
There are 650,000 people in a city. Every 15 minutes, the local radio and television stations broadcast a tornado warning. During each 15-minute time period, 42% of the people who had not yet heard the warning become aware of the approaching tornado. How many people have heard the news? a. After 1
Suppose you invest $500 in a bank that pays 5.5% annual interest compounded quarterly. a. How much money will you have after five years? b. Suppose you also deposit an additional $150 at the end of every three months. How much will you have after five years?
Consider the explicit formula un = 81(1/3)n-1 a. List the first six terms, u1 to u6. b. Write a recursive formula for the sequence.
Find the exact value for a. cos 15° b. cos 75°a
Find S1, S2, S3, S4, and S5 for this sequence: 2, 6, 10, 14, 18.
Consider the rational equation y = 4x + 3 / 2x - 1. a. Rewrite the equation as a transformation of the parent function y = 1/x. b. What are the asymptotes of y = 4x + 3 / 2x - 1? c. The point (1, 1) is on the graph of the parent function. What is its image on the transformed function?
Write each expression as a sum of terms, then calculate the sum.a.b.
Find the sum of the first 50 multiples of 6: {6, 12, 18, . . ., u50}.
Find the sum of the first 75 even numbers, starting with 2.
Find these values.a. Find u75 if un = 2n - 1.b. Findc. Find
Consider the graph of the arithmetic sequence shown at right.a. What is the 46th term?b. Write a formula for un.c. Find the sum of the heights from the horizontal axis of the first 46 points of the sequence's graph.
Suppose you practice the piano 45 min on the first day of the semester and increase your practice time by 5 min each day. How much total time will you devote to practicing duringa. The first 15 days of the semester? b. The first 35 days of the semester?
Jessica arranges a display of soup cans as shown.a. List the number of cans in the top row, the second row, the third row, and so on, down to the tenth row. b. Write a recursive formula for the terms of the sequence in 9a. c. If the cans are to be stacked 47 rows high, how many cans will it take to
Consider the repeating decimal 0.444..., or 0.4. a. Express this decimal as the sum of terms of an infinite geometric series. b. Identify the first term and the ratio. c. Use the formula you learned in this lesson to express the sum as a ratio of integers.
A flea jumps ft right, then ft left, then ft right, and so on. To what point is the flea zooming in?
Suppose square ABCD with side length 8 in. is cut out of paper. Another square, EFGH, is placed with its corners at the midpoints of ABCD. A third square is placed with its corners at midpoints of EFGH, and so on.a. What is the perimeter of the tenth square?b. What is the area of the tenth
The fractal known as the SierpiÅ„ski triangle begins as an equilateral triangle with side length 1 unit and area square units. The fractal is created recursively by replacing the triangle with three smaller congruent equilateral triangles such that each smaller triangle shares a vertex $The fractal known as the SierpiÅ„ski triangle begins as an$
Match each equation to a graph.A. y = 10(0.8)xB. y = 10 - 10(0.75)xC. y = 3 + 7(0.7)xD. y = 10 - 7(0.65)xi.ii. iii. iv.
A computer software company decides to set aside $100,000 to develop a new video game. It estimates that development will cost $955 the first week and that expenses will increase by $65 each week. a. After 25 weeks, how much of the development budget will be left? b. How long can the company keep
Hans sees a dog. The dog has four puppies. Four cats follow each puppy. Each cat has four kittens. Four mice follow each kitten. How many legs does Hans see? Express your answer using sigma notation.
Repeat the three parts of Exercise 1 with the repeating decimal 0.474747..., or 0.47.
Repeat the three parts of Exercise 1 with the repeating decimal 0.123123123..., or 0.123.
An infinite geometric sequence has a first term of 20 and a sum of 400. What are the first five terms?
An infinite geometric sequence contains the consecutive terms 128, 32, 8, and 2. The sum of the series is 43,690.6. What is the first term?
Consider the sequence u1 = 47 and un = 0.8un - 1 where ‰¥ 2. Finda.b. c. d.
Consider the sequence un = 96(0.25)n - 1.a. List the first ten terms, u1 to u10.b. Find the sumc. Make a graph of partial sums for 1 ‰¤ n ‰¤ 10. d. Find the sum
A ball is dropped from an initial height of 100 cm. The rebound heights to the nearest centimeter are 80, 64, 51, 41, and so on. What is the total distance the ball will travel, both up and down?
A sporting event is to be held at the Superdome in New Orleans, Louisiana, which holds about 95,000 people. Suppose 50,000 visitors arrive in New Orleans and spend $500 each. In the month after the event, the people in New Orleans spend 60% of the income from the visitors. The next month, 60% is
For each partial sum equation, identify the first term, the ratio, and the number of terms.a.b. c. d. - 40 + 40(2.5)6 = 9725.625
As a contest winner, you are given the choice of two prizes. The first choice awards $1000 the first hour, $2000 the second hour, $3000 the third hour, and so on. For one entire year, you will be given $1000 more each hour than you were given during the previous hour. The second choice awards 1 the
Consider the geometric series 5 + 10 + 20 + 40 + ··· a. Find the first seven partial sums, S1, S2, S3, ... , S7. b. Do the partial sums create a geometric sequence? c. If u1 is 5, find value(s) of r such that the partial sums form a geometric sequence.
Consider the seriesa. Is this series arithmetic, geometric, or neither? b. Find the sum of this series.
List terms to finda.b.
The 32 members of the Greeley High chess team are going to have a tournament. They need to decide whether to have a round-robin tournament or an elimination tournament. (Read the Recreation Connection.) a. If the tournament is round-robin, how many games need to be scheduled? b. If it is an
What monthly payment is required to pay off an $80,000 mortgage at 8.9% interest in 30 years?
Develop the parametric equations of a hyperbola by following these steps. a. Write the equation of a unit hyperbola in standard form using x and y. b. Replace x with 1 / cos t and solve for y in terms of t. You should get a square root equation. c. Add the terms under the radical by finding a
The Magic Garden Seed Catalog advertises a bean with unlimited growth. It guarantees that with proper watering, the bean will grow 6 in. the first week and the height increase each subsequent week will be three-fourths of the previous week's height increase. "Pretty soon," the catalog claims, "Your
Write the polynomial equation of least degree that has integer coefficients and zeros -3 + 2i and 2/3.
Consider the geometric sequence 256, 192, 144, 108, . . . a. What is the eighth term? b. Which term is the first one smaller than 20? c. Find u7. d. Find S7.
Find each partial sum of this sequence. u1 = 40 un = 0.6un - 1 where n ≥ 2 a. S5 b. S15 c. S25
Identify the first term and the common ratio or common difference of each series.Then find the partial sum.a. 3.2 + 4.25 + 5.3 + 6.35 + 7.4b. 3.2 + 4.8 + 7.2 + ··· + 36.45c.d.
Find the missing value in each set of numbers. a. u1 = 3, r = 2, S10 = ? b. u1= 4, r = 0.6, S? ≈ 9.999868378 c. u1 = ?, r = 1.4, S15 ≈1081.976669 d. u1= 5.5, r = ?, S18 ≈ 66.30642497
Find the nearest integer value for n if 3.2(1-08n / 1-0.8) is approximately 15.
Consider the sequence u1 = 8 and un = 0.5un - 1 where n ‰¥ 2. Finda.b. c. d. Explain what is happening to these partial sums as you add more terms.
Suppose you begin a job with an annual salary of $17,500. Each year, you can expect a 4.2% raise. a. What is your salary in the tenth year after you start the job? b. What is the total amount you earn in ten years? c. How long must you work at this job before your total earnings exceed $1 million?
An Indian folktale, recounted by Arab historian and geographer Ahmad al-Yaqubi in the 9th century, begins, "It is related by the wise men of India that when Husiya, the daughter of Balhait, was queen . . . ," and goes on to tell how the game of chess was invented. The queen was so delighted with
According to the table values, what is the sum of your infinite series? Use the explicit formula S = u1/1-r to confirm this sum.
Consider the arithmetic sequence: 3, 7, 11, 15, ... a. What is the 128th term? b. Which term has the value 159? c. Find u20. d. Find S20.
Consider the geometric sequence: 100, 84, 70.56, ...a. Which term is the first one smaller than 20?b. Find the sum of all the terms that are greater than 20.c. Find the value ofd. What happens to Sn as n gets very large?
Given plenty of food and space, a particular bug species will reproduce geometrically, with each pair hatching 24 young at age 5 days. (Assume that half the newborn bugs are male and half female, and are ready to reproduce in five days. Also assume each female bug can only reproduce once.)
Consider the series 125.3 + 118.5 + 111.7 + 104.9 + ··· a. Find S67. b. Write an expression for S67 using sigma notation.
Emma's golf ball lies 12 ft from the last hole on the golf course. She putts and, unfortunately, the ball rolls to the other side of the hole, 2/3 as far away as it was before. On her next putt, the same thing happens. a. If this pattern continues, how far will her ball travel in seven putts? b.
A flea jumps ½ ft, then ¼ ft, then 1/8 ft, and so on. It always jumps to the right. a. Do the jump lengths form an arithmetic or geometric sequence? What is the common difference or common ratio? b. How long is the flea's eighth jump, and how far is the flea from its starting point? c. How long
For 7a-c, use u1= 4. Round your answers to the nearest thousandth. a. For a geometric series with r = 0.7, find S10 and S40. b. For r = 1.3, find S10 and S40. c. For r = 1, find S10 and S40. d. Graph the partial sums of the series in 7a-c. e. For which value of r (0.7, 1.3, or 1) do you have a
Consider the series 0.8 + 0.08 + 0.008 + ··· a. Find S10. b. Find S15. c. Express the sum of infinitely many terms as a ratio of integers.
You know how to write the equation of a continuous function that passes through the discrete points of a sequence, (n, un). For example, the function y = 200(0.8)x - 1 passes through the sequence of points representing the distance in inches that a ball falls on each bounce. Write a continuous
The explicit formula for a partial sum of a geometric series is Sn = u1(1 -rn) / 1 - r. To find the sum of an infinite geometric series, you can imagine substituting for n. Explain what happens to the expression u1(1 -rn) / 1 - r when you do this substitution.
Since Chapter 1, you have solved problems about monthly payments, such as auto loans and home mortgages. You've learned how to find the monthly payment, P, required to pay off an initial amount, A0, over n months with a monthly percentage rate, r. With series, you can find an explicit formula to
Nina has observed that her coach does not coordinate the color of his socks to anything else that he wears. Guessing that the color is a random selection, she records these data during three weeks of observation: black, white, black, white, black, white, black, red, white, red, white, white, white,
Find the number of equally likely outcomes of each event described for a two-die roll. Then write the probability of each event. a. The dice sum to 9. b. The dice sum to 6. c. The dice have a difference of 1. d. The sum of the dice is 6, and their difference is 2. e. The sum of the dice is at most
Consider the diagram at right.a. What is the total area of the square? b. What is the area of the shaded region? c. Suppose the horizontal and vertical coordinates are randomly chosen numbers between 0 and 12, inclusive. Over the long run, what ratio of these points will be in the shaded area? d.
Suppose x and y are both randomly chosen numbers between 0 and 8. (The numbers are not necessarily integers.) a. Write a symbolic statement describing the event that the sum of the two numbers is at most 6. b. Draw a two-dimensional picture of all possible outcomes, and shade the region described
Use the histogram at right for 13a-d.a. Approximate the frequency of scores between 80 and 90. b. Approximate the sum of all the frequencies. c. Find P(a score between 80 and 90). d. Find P(a score that is not between 80 and 90).
A 6 in. cube painted on the outside is cut into 27 smaller congruent cubes. Find the probability that one of the smaller cubes, picked at random, will have the specified number of painted faces. a. Exactly one b. Exactly two c. Exactly three d. No painted face
Where, in atoms, do electrons reside? The graph at right shows the probability of the electron of a hydrogen atom being at various distances from the nucleus at any given moment. The distances are measured in picometers (pm). A picometer is 1 10 - 12 m. Use the graph to answer these questions.a. At
Write log a - log b + 2 log c as a single logarithmic expression.
Consider this system of inequalities:a. Graph the triangle defined by this system. b. Give the coordinates of the vertices of the triangle in 19a. c. Find the area of the triangle in 19a.
This table shows numbers of students in several categories at Ridgeway High. Find the probabilities described below. Express each answer to the nearest 0.001.a. What is the probability that a randomly chosen student is female? b. What is the probability that a randomly chosen student is an 11th
Consider these two sets of data. i. {5, 23, 36, 48, 63} ii. {112, 115, 118, 119, 121} a. Which set would you expect to have the larger standard deviation? Explain your reasoning. b. Calculate the mean and the standard deviation of each set. c. Predict how the mean and the standard deviation of each
The graph of the shaded area at right shows all possible combinations of two numbers, x and y. Use the graph and basic area formulas to answer each question. Express each answer to the nearest 0.001.a. What is the probability that x is between 0 and 2? b. What is the probability that y is between 0
Find each probability. a. Each day, your teacher randomly calls on 5 students in your class of 30. What is the probability you will be called on today? b. If 2.5% of the items produced by a particular machine are defective, then what is the probability that a randomly selected item will not be
Rank i-iii according to the method that will best produce a random integer from 0 to 9. Support your reasoning with complete statements. i. The number of heads when you drop nine pennies ii. The length, to the nearest inch, of a standard 9 in. pencil belonging to the next person you meet who has a
a. What do you think the long-run experimental probability will be? b. Make a graph of the cumulative ratio of 3's versus the number of tosses. Plot the points (cumulative number of tosses, cumulative ratio of 3's). Then plot three more points as you extend the domain of the graph to 2400, 3600,
Consider rolling a green die and a white die. The roll (1, 5) is different from (5, 1). a. How many different outcomes are possible for this two-die experiment? b. How many different outcomes are possible in which there is a 4 on the green die? Draw a diagram to show the location of these points.
Name two different ways to generate random numbers from 0 to 10.
Find the term specified for each binomial expansion. a. the first term of (1 + x / 12)99 b. the last term of (1 + x / 12)99 c. the tenth term of (a + b)21
Suppose you roll two octahedral (eight-sided) dice, numbered 1-8. a. Draw a diagram that shows all possible outcomes of this experiment. b. Indicate on your diagram all the possible outcomes for which the sum of the dice is less than 6. c. What is the probability that the sum is less than 6? d.
Answer each geometric probability problem.a. What is the probability that a randomly plotted point will land in the shaded region pictured at right?b. One thousand points are randomly plotted in the rectangular region shown below. Suppose that 374 of the points land in the shaded portion of the
a. Draw a tree diagram representing all of the possible results. (Assume all five questions are answered.) b. How many possible ways are there of getting three true and two false answers? c. How could you use combinations or permutations to answer 4b? d. Suppose you are sure that the answers to the

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