HEART OF MATHEMATICS
4th Edition
ISBN: 9781119760061
Author: Burger
Publisher: WILEY
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Textbook Question
Chapter 7.3, Problem 5MS
Divining dimension. If a fractal object requires
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There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and
use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three
investment?
STEP 1: The formula for compound interest is
A =
nt
= P(1 + − − ) n²,
where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to
A = Pert
Find r and n for each model, and use these values to write A in terms of t for each case.
Annual Model
r=0.10
A = Y(t) = 1150 (1.10)*
n = 1
Quarterly Model
r = 0.10
n = 4
A = Q(t) = 1150(1.025) 4t
Continuous Model
r=0.10
A = C(t) =…
The following ordered data list shows the data speeds for cell phones used by a
telephone company at an airport:
A. Calculate the Measures of Central Tendency from the ungrouped data list.
B. Group the data in an appropriate frequency table.
C. Calculate the Measures of Central Tendency using the table in point B.
D. Are there differences in the measurements obtained in A and C? Why (give at
least one justified reason)?
I leave the answers to A and B to resolve the remaining two.
0.8
1.4
1.8
1.9
3.2
3.6
4.5
4.5
4.6
6.2
6.5
7.7
7.9
9.9
10.2
10.3
10.9
11.1
11.1
11.6
11.8
12.0
13.1
13.5
13.7
14.1
14.2
14.7
15.0
15.1
15.5
15.8
16.0
17.5
18.2
20.2
21.1
21.5
22.2
22.4
23.1
24.5
25.7
28.5
34.6
38.5
43.0
55.6
71.3
77.8
A. Measures of Central Tendency
We are to calculate:
Mean, Median, Mode
The data (already ordered) is:
0.8, 1.4, 1.8, 1.9, 3.2, 3.6, 4.5, 4.5, 4.6, 6.2, 6.5, 7.7, 7.9, 9.9, 10.2, 10.3, 10.9,
11.1, 11.1, 11.6,
11.8, 12.0, 13.1, 13.5, 13.7, 14.1, 14.2, 14.7, 15.0, 15.1, 15.5,…
A tournament is a complete directed graph, for each pair of vertices x, y either (x, y) is an arc or
(y, x) is an arc. One can think of this as a round robin tournament, where the vertices represent
teams, each pair plays exactly once, with the direction of the arc indicating which team wins.
(a) Prove that every tournament has a direct Hamiltonian path. That is a labeling of the teams
V1, V2,..., Un so that vi beats Vi+1. That is a labeling so that team 1 beats team 2, team 2
beats team 3, etc.
(b) A digraph is strongly connected if there is a directed path from any vertex to any other
vertex. Equivalently, there is no partition of the teams into groups A, B so that every team
in A beats every team in B. Prove that every strongly connected tournament has a directed
Hamiltonian cycle. Use this to show that for any team there is an ordering as in part (a) for
which the given team is first.
(c) A king in a tournament is a vertex such that there is a direct path of length at most 2 to
any…
Chapter 7 Solutions
HEART OF MATHEMATICS
Ch. 7.1 - The incredible shrinking duck. On the Quacked...Ch. 7.1 - Multiplicity (S). In the Sierpinski Triangle,...Ch. 7.1 - Different sizes. In the fern, find three reduced...Ch. 7.1 - Blooming broccoli. In the bunch of broccoli, find...Ch. 7.1 - Prob. 5MSCh. 7.1 - Prob. 6MSCh. 7.1 - Maybe moon. What features of the fractal forgeries...Ch. 7.1 - Exposing forgeries. What features of the fractal...Ch. 7.1 - Natures way. Find some examples of self-similarity...Ch. 7.1 - Do it yourself (H). Draw a figure that contains...
Ch. 7.1 - With a group of folks. In a small group, discuss...Ch. 7.1 - Prob. 12MSCh. 7.2 - A search for self. What does self-similarity mean?Ch. 7.2 - Desperately seeking similarity. Which of these...Ch. 7.2 - Too many triangles? At stage 0, the Sierpinski...Ch. 7.2 - Counting Koch. Look at the early stages of the...Ch. 7.2 - Argyle art. Take four line segments and place them...Ch. 7.2 - Natures way. Find several examples of objects in...Ch. 7.2 - Whos the fairest? Can you position three mirrors...Ch. 7.2 - Billiards and mirrors. On an idealized, square...Ch. 7.2 - MTV. Youve become a rock star and consequently...Ch. 7.2 - Photo op. Suppose you arrange two mirrors facing...Ch. 7.2 - How many mes? Arrange mirrors and camera as in...Ch. 7.2 - Quacker, Quacker, Quacker. Suppose you had the job...Ch. 7.2 - Sierpinski hexed (S). Take an equilateral triangle...Ch. 7.2 - The Kinks (ExH). Kochs kinky curve is created by...Ch. 7.2 - Four times (H). Draw a picture on a square piece...Ch. 7.2 - Burger heaven (S). Sketch a picture of a...Ch. 7.2 - Ice cream cones. Draw a picture of an ice cream...Ch. 7.2 - Sierpinski boundary. Take the boundary of a...Ch. 7.2 - Catching Zs (H). Take a Z. Put in nine smaller Zs,...Ch. 7.2 - Replacement pinwheel. Take a 1,2,2 right triangle....Ch. 7.2 - Koch Stool (ExH). Start with a line segment, mark...Ch. 7.2 - Koch collage stool. Given the Koch Stool described...Ch. 7.2 - Sierpinski shooting. Suppose that you were playing...Ch. 7.2 - Sierpinski target practice (H). What sequence of...Ch. 7.2 - Cantor Set. Start with the interval [0, 1]. Build...Ch. 7.2 - Cantor luck (H). Start with the point 0. Flip a...Ch. 7.2 - Cantor Square (S). Take a square. Make four...Ch. 7.2 - Cantor Square shrunk. Take the Cantor Square....Ch. 7.2 - Cantor Squared. Draw the four corners of a square,...Ch. 7.2 - Hexed again. Suppose you start with the three...Ch. 7.2 - Prob. 31MSCh. 7.2 - Antoines necklace. Antoines necklace is a delicate...Ch. 7.2 - Menger jacks. For the game of jacks, lets imagine...Ch. 7.2 - A tighter weave. Look back at the Tight Weave...Ch. 7.2 - A looser weave. Lets modify the carpet-designing...Ch. 7.2 - From where? Look at this fractal. What collage...Ch. 7.2 - Treed. Describe collage-making instructions that...Ch. 7.2 - Flaky (H). Describe collage making instructions...Ch. 7.2 - How big a hole? In the Tighter weave (34) and...Ch. 7.2 - 4D fractal. Describe a fractal in 4-dimensional...Ch. 7.2 - Personal perspectives. Write a short essay...Ch. 7.2 - With a group of folks. In a small group, discuss...Ch. 7.2 - Graphing Koch. Plot the following points in the...Ch. 7.2 - Koch adds life. Look at the points given in the...Ch. 7.2 - White out. Each stage of the Sierpinski triangle...Ch. 7.2 - Measuring the length of your Koch. The Koch stool...Ch. 7.2 - Counting Koch segments (H). Suppose you start with...Ch. 7.3 - Parallel grams. At right we see a parallelogram....Ch. 7.3 - Moving on up. Heres a triangle along with some...Ch. 7.3 - Bigger rug. Heres a picture of the Sierpinski...Ch. 7.3 - Dimension connection. What equation relates the...Ch. 7.3 - Divining dimension. If a fractal object requires...Ch. 7.3 - Stay inbounds. Give two consecutive integers that...Ch. 7.3 - Regular things (H). Find the fractal dimension of...Ch. 7.3 - More regular things. Find the fractal dimension of...Ch. 7.3 - Any right triangle. Take any right triangle. It...Ch. 7.3 - Sierpinski carpet (S). Compute the fractal...Ch. 7.3 - Koch Stool. Compute the fractal dimension of the...Ch. 7.3 - Cantor Set (H). The Cantor Set was constructed by...Ch. 7.3 - Cantor reduced. Suppose you take a unit interval,...Ch. 7.3 - Long Koch (ExH). The first stage in the...Ch. 7.3 - Pius (H). This fractal plus sign is self-similar...Ch. 7.3 - Tiniler triangles (S). Suppose you make something...Ch. 7.3 - Menger Sponge (ExH). Compute the fractal dimension...Ch. 7.3 - Thinning. Take a square. Make two copies, each...Ch. 7.3 - Not much. What is the fractal dimension of the...Ch. 7.3 - Koched (H). Create a Koch-like curve with fractal...Ch. 7.3 - Find a fractal. Describe a fractal having...Ch. 7.3 - Prob. 22MSCh. 7.3 - Personal perspectives. Write a short essay...Ch. 7.3 - With a group of folks. In a small group, discuss...Ch. 7.3 - Prob. 25MSCh. 7.3 - Power beyond the mathematics. Provide several...Ch. 7.3 - Varying dimensions. Using any method, including...Ch. 7.3 - Dimensional thinking. The text defines the...Ch. 7.3 - Power play (H). Using any method, including...Ch. 7.3 - Marching madness. The marching band wants to...Ch. 7.3 - More marching madness. For the second drill in the...Ch. 7.4 - Not Raul. Who was Gaston Julia?Ch. 7.4 - Use your imagination. What does the number i...Ch. 7.4 - Complex plots. The complex number 3+2i is plotted...Ch. 7.4 - Prob. 4MSCh. 7.4 - Outie or innie? If you use the point a+bi to build...Ch. 7.4 - Arithmetic. Compute the following sums and...Ch. 7.4 - More arithmetic (ExH). Compute the following sums...Ch. 7.4 - Quick draw. Two complex numbers are marked and...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - 1III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - Orange Julias. Pictured here are some Julia Sets...Ch. 7.4 - Julia Webbed. Visit the Heart of Mathematics Web...Ch. 7.4 - Great escape? For each of the complex...Ch. 7.4 - Great escape? For each of the complex numbers...Ch. 7.4 - Great escape? For each of the complex numbers...Ch. 7.4 - Mandelbrot or not? (H). The images on the...Ch. 7.4 - Mandelbrot or not? (H). The images on the...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Mandelbrot origins. Using your insights from...Ch. 7.4 - Prob. 36MSCh. 7.4 - Bounded Julia (ExH). Why is no Julia Set...Ch. 7.4 - Prisoner. Find a complex number a+bi that is fixed...Ch. 7.4 - Mandelbrot connections. Can every two points on...Ch. 7.4 - Personal perspectives. Write a short essay...Ch. 7.4 - With a group of folks. In a small group, discuss...Ch. 7.4 - Power beyond the mathematics. Provide several...Ch. 7.4 - Quadratic complexity (H). The quadratic formula...Ch. 7.4 - Special products. Compute the following products...Ch. 7.4 - Somethings fishy. The treasurer of the Marine...Ch. 7.4 - On the line. In the xy-plane, sketch the graph of...Ch. 7.4 - A line with a complex. Plot the points in the...Ch. 7.5 - Any interest? Suppose you put $1000 in a bank...Ch. 7.5 - Urban expansion. In the year 2004 the village of...Ch. 7.5 - Pre-sushi. Suppose a pond can sustain a maximum of...Ch. 7.5 - A booby trap. A small landmass off the Galapagos...Ch. 7.5 - Too many. The island of Birdburg has an area of...Ch. 7.5 - Call your shots. For each picture below, trace the...Ch. 7.5 - Getting cornered. Starting from the spot indicated...Ch. 7.5 - Double your money (H). Suppose you I deposit $1000...Ch. 7.5 - Too many(S). Earths surface is approximately...Ch. 7.5 - Rice bowl (H). One day long ago, the Emperor of...Ch. 7.5 - Natures way. Find some examples, other than those...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Explosion. Devise a new initial population in the...Ch. 7.5 - Extinction. Construct a new initial population in...Ch. 7.5 - Periodic population (H). Construct a new initial...Ch. 7.5 - Programmed population. On a computer or...Ch. 7.5 - Programmed population: the next generation. Using...Ch. 7.5 - How many now? Suppose that a population is modeled...Ch. 7.5 - Fibonacci. Fibonacci numbers are constructed using...Ch. 7.5 - Fibonacci again. For each of the two columns you...Ch. 7.5 - Prob. 29MSCh. 7.5 - Cobweb plots (ExH). Here we visualize the...Ch. 7.5 - More spiders (S). Here is the graph of...Ch. 7.5 - Arachnids. Here is the graph of y=4x(1x). Start on...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - More cobwebs (H). Consider the inverted V graph...Ch. 7.5 - Yet more cobwebs. Given the inverted V graph...Ch. 7.5 - Cantors cuts. Start with the unit interval [0, 1]....Ch. 7.5 - How much ¡s gone? In the construction of the...Ch. 7.5 - How much remains? Consider numbers that are not in...Ch. 7.5 - Personal perspectives. Write a short essay...Ch. 7.5 - With a group of folks. In a small group, discuss...Ch. 7.5 - Prob. 43MSCh. 7.5 - Power beyond the mathematics. Provide several...Ch. 7.5 - Life happens. A particular starting configuration...Ch. 7.5 - Perplexing population predictions. Verhulst model...Ch. 7.5 - More prediction peril. Consider the previous...Ch. 7.5 - Cobweb parabola. Mindscapes 3032 describe cobweb...Ch. 7.5 - Two sad and one happy parabolas. Building on your...Ch. 7.6 - Does this thing come with a warranty? If you use...Ch. 7.6 - Root repeater. Find a calculator with a square...Ch. 7.6 - Transforming experience. The equation y=4x(1x) is...Ch. 7.6 - Up and over. In this graph, we see a red diagonal...Ch. 7.6 - Over and up. Here is a simpler version of the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - Too high. Consider a tent function that got too...Ch. 7.6 - More gone. In the too high tent function in the...Ch. 7.6 - Too short. Consider a short tent function. Will...Ch. 7.6 - Where to? Using the transformation y=3.5x(1x),...Ch. 7.6 - Calculator slips. Using the logistic...Ch. 7.6 - Take stock. Pick a stock. Some Web sites will...Ch. 7.6 - Repulsive. If you take the number 1 and square it,...Ch. 7.6 - Attractive. Again consider the process of repeated...Ch. 7.6 - Sierpinski attractor. Remember how the Sierpinski...Ch. 7.6 - Two step. Consider the transformation that takes...Ch. 7.6 - Periodic attraction (S). Again consider the...Ch. 7.6 - Periodic attraction. Again consider the...Ch. 7.6 - Four-peat (H). Consider the equation y=3.5x(1x)....Ch. 7.6 - Nearly fourly. Consider the equation y=3.5x(1x)....Ch. 7.6 - Tent attraction? Consider the point 0.4, which is...Ch. 7.6 - Becoming periodic. The point 0.4 is of period 2 in...Ch. 7.6 - The Earth moved (ExH). Consider a transformation...Ch. 7.6 - Poles apart. Consider the same transformation as...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - With a group of folks. In a small group, discuss...Ch. 7.6 - Power beyond the mathematics. Provide several...Ch. 7.6 - V-quation. Look at the graph for Mindscape 4. The...Ch. 7.6 - Meeting up. Look at the graph for Mindscape 38....Ch. 7.6 - We meet again. Look at the graph for Mindscape 39....Ch. 7.6 - Calculating percentages (H). Return to Table 1 on...Ch. 7.6 - Calculating percentages II. Repeat Mindscape 48...
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