Advanced Algebra
Chapter 4 Answers
Patterns and Structure in Algebra
Arithmetic sequences and series can be used to analyze patterns in nature as diverse as orbits of comets and diameters of trees.
Part A, Arithmetic Sequences
1. Like planets, "periodic" comets rotate in well-defined orbits about the Sun. The length of time it takes a comet to complete one rotation about the Sun is called the comet's period. Find data on the first 50 periodic comets.
a. Find the period of Comet Olbers (#13P). Periods are given in years. [69.56 years]
b. Find the years of Comet Olbers's successive returns to the position it occupied at the time of its discovery, from the year of its discovery to the year of its next expected sighting. [1815.00, 1884.56, 1954.12, 2023.68]
c. How do you know that the above terms form an arithmetic sequence? [The years of consecutive returns differ by the period, 69.56 years, which is the common difference of the sequence.]
d. Historical records show that a bright object was visible in the night sky throughout the world in the year 1682. Might the object have been Comet Olbers? Explain. [No. Subtraction of successive periods of 69.56 years from 1815 shows that Comet Olbers's two previous appearances before its discovery were in 1745 and 1676. Students may be interested to know that the object was Halley's Comet.]
Part B, Arithmetic Series
2. Each year a tree trunk increases its diameter through the addition of a "ring" of new material. If weather, nutrition, and other growing conditions do not change, ring widths will remain fairly constant. If conditions change, ring widths will vary from year to year.
The figure below shows a cross-section of a 5-year-old tree that has grown through the addition of rings of constant width.
a. At the Tree Ring Growth Web site, find the mean ring width of "Tree 1" for the 10 years 1985-1994. Round to the nearest hundredth. [2.15 mm]
b. Assume that the value you calculated in 2a above gives the tree's radius at the end of Year 1, as well as the width of each new annual ring.Write an arithmetic sequence expressing annual increases in the tree's radius. What is the common difference? [2.15, 2.15, 2.15, 2.15,... ; The common difference is 0.]
c. Find the first five partial sums of the series based on the above sequence. What do the sums represent? [2.15 mm, 4.30 mm, 6.45 mm, 8.60 mm, 10.75 mm; The sums represent the radii of the tree at the end of every year from Year 1 through Year 5.]
d. Find the area of the trunk's cross-section after Year 1. Then find the cross-sectional area that is added to the trunk during each of the next four years. Write areas in terms of π. How do you know that the sequence is arithmetic? [4.6225π, 13.8675π, 23.1125π, 32.3575π, 41.6025π; The sequence is arithmetic because there is a common difference of 9.245π between terms.]
e. Use sigma notation to express the trunk's cross-sectional area after five years. What is the area?
Geometric sequences and series have a variety of applications, from cell division to consumer fraud to the problem of whether a snail can complete a long and perilous journey!
Part A, Geometric Sequences
1. Read about mitosis, the most common type of cell division in plant and animal cells.
a. Complete the table to show how many cells exist after each of the first ten divisions.
| divisions | 0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| cells | 1 |
| divisions | 0 | 1 |
2 | 3 | 4 |
5 |
6 |
7 |
8 |
9 |
10 |
| cells | 1 |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
b. Do the values in Row 2 of the table form a geometric sequence? Explain. [Yes. Each term can be found by multiplying the previous term by the common ratio, 2.]
c. Write an expression the 100 th term of the sequence. [299]
Part B, Geometric Series
2. "Pyramid schemes" are illegal for good reason: they claim that people who join the scheme will make large amounts of money, when in fact those people may well lose money. Read about pyramid schemes.
a. One common pyramid scheme is a chain letter. Read Mr. Smith's letter below. If Mr. Smith follows the directions on the chain letter, and if all the people on the list follow the directions, how much money will Mr. Smith receive after the first round of the letter? [$10]
Hello Mr. Smith!
Below is a list of 5 people. Send $1 to each person on the list. Then cross the first name off the list, add your name to the bottom of the list, and send the letter to 10 of your friends.
1. Mr. Henson, 111 Main St., Chicago IL 26519
2. Mrs. Jones, 5 Maple Lane, Finnegan OH 75123
3. Mr. Mason, 44 Oak St., Beatty, UT 33221
4. Mrs. Klein, 502 Maple, Erie, PA 77091
5. Mr. Webb, 51 Pine St., Detroit, MI 98765
b. If everyone continues to follow directions, what is the total amount of money Mr. Smith will have received after the second round? after the third round? [$110; $1110]
c. Suppose that everyone continues to follow directions. Write a geometric series expressing the total amount Mr. Smith will have received after the n th round. [$10 + $100 + $1000 + ... +$10n]
d. Find the total amount Mr. Smith will have received after the 8 th round. [S10 = {10 (1 108)} / (1 10) = $111,111,110]
e. Why, in all likelihood, will Mr. Smith not receive the amount you calculated above? [To receive $111,111,110, each of 111,111,110 people will have to send $1 to Mr. Smith. Long before this happens--typically in the first or second round--some people will break the chain, failing to contact more people or to pay the money requested of them. At this point, the pyramid will collapse. Instead of $111,111,110, Mr. Smith will be lucky to get his $5 back.]
Part C, Investigating the Concept of a Limit
4. Read about Zeno's Paradoxes.
a. A snail set out to travel a distance of 1 yard. In the first stage of its journey it traveled 1/2 yard. In Stage 2 it traveled half the remaining distance. It continued in this fashion, at each stage traveling half the remaining distance. Write a geometric series giving the distances the snail traveled during each of the first five stages of the journey. [1/2 yd + 1/4 yd + 1/8) yd + 1/16 yd + 1/32 yd]
b. What is the common ratio of the series? [1/2]
c. What is the sum of the infinite series which has the above terms as its first five terms? Is the sum equal to the distance the snail must travel?
d. Explain Zeno's Paradox in relation to this problem. [Zeno would say: First the snail must travel half the total distance. Then it must travel half the remaining distance. Then it must travel half the remaining distance. And so on. The halves will get smaller and smaller, to be sure, but there will always be another half to be negotiated, and in front of that another. Clearly, the snail will never reach its destination.]
Square roots of positive numbers have wide applications in science. Square roots of negative numbers lead to the new and exciting field of fractal geometry. You'll explore both types of square roots below.
Part B, Rational and Irrational Numbers
1. Refer to the picture below. An object propelled upward at low initial
velocity from a point A on a planet's surface will fall back to the surface,
point B. If the initial velocity is increased sufficiently, the object will
still fall, point C, but its descent path will exactly match the curvature
of the planet. Such an object will go into orbit around the planet. The
initial velocity required to achieve orbit is called orbital velocity.
If the initial velocity is increased even more, the object will escape from the planet's gravitional field altogether, point D. The required initial velocity is called escape velocity. Escape velocity v (km/sec) for a planet of mass m (kg) and radius r (km) is given by
v = 0.000 000 000 37
a. Find the mass and radius of the earth and the moon. (Masses are written for easy input into a scientific calculator. To input 4.6e8, press
. Instead of
, your calculator may have
. [earth: 5.97e24 kg, 6378 km; moon: 7.35e22 kg, 1738 km]
b. Find the velocity that was necessary for NASA Apollo missions to escape the earth's gravity in order to travel to the moon. [11.32 km/sec]
c. What escape velocity did astronauts on the moon have to achieve in order to return to earth? [2.41 km/sec]
d. Jupiter's mass is about 25,000 times the mass of the moon. Jupiter's radius is about 40 times the radius of the moon. Show how you can estimate Jupiter's escape velocity in comparison with that of the moon, without actually calculating Jupiter's escape velocity.
Part D, Graphing Complex Numbers
2a. Let f (z) = z2 0.4 + 0.2i. Find f (z) for z = 1 + 1.6i. [1.96 + 3.4i]
b. Find f (z) for z equal to the value of f (z) that you found above. [7.7184 13.328i]
c. In 2a and 2b, you found two iterates of a complex function. If you were to graph many such iterates, you would create a design of gradually increasing intricacy. Computers can graph huge numbers of iterates almost instantly. The results are the beautiful and complex designs we know as fractal images. See "Mandelbrot Explorer," an interactive fractal image that you can explore in depth. Notice that the size of the complex plane shown is indicated below the image: the real axis x extends from 1 to +0.5, the imaginary axis y from 1.25 to +1.25.
d. Choose a zoom factor of "ZoomIn X 4." Click on the fractal image at a point you would like to explore in more depth. In a few seconds the computer will show you the portion of the image you clicked on, now at 4 times magnification. Notice that the new endpoints of the real and imaginary axes are given below the image.
e. Choose a zoom-in factor and click on a new point. Continue in this manner, displaying a portion of the image at increasing levels of magnification. In general terms, describe what you see. [Descriptions will vary. Students should note that instead of growing simpler, fractal images exhibit beautiful new complexities at each stage of magnification. They may also note that the shapes they observe, while not congruent, have a decided resemblance to one another. This is the fractal property known as self-similarity.]
f. Click on Mandelbrot Explorer Gallery to view other fractal images that have been created using the "Mandelbrot Explorer."