Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
7th Edition
ISBN: 9780134705187
Author: Jeffrey O. Bennett, William L. Briggs
Publisher: PEARSON
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Textbook Question
Chapter 8.A, Problem 25E
Bacteria in a Bottle Parable. Use the bacteria parable presented in the text.
23. How many bacteria are in the bottle at 11:50? What traction of the bottle is full at that time?
- How many bacteria are in the bottle at 11:15? What traction of the bottle is full at that time?
- Knee-Deep in Bacteria. The total surface area of Earth is about 3.1 X 1014 m2 Assume that the bacteria continued their doublings for a total of two hours (as discussed in the text), at which point they were distributed uniformly over Earth's surface. How deep would the bacterial layer be? Would it be knee-deep, mote than knee-deep, or less than knee-deep? (Hint: You can find the approximate depth by dividing the bacteria volume by (Earth's surface area.)
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Students have asked these similar questions
Module Code: MATH380202
3. (a) Let {} be a white noise process with variance σ2.
Define an ARMA(p,q) process {X} in terms of {+} and state (without proof)
conditions for {X} to be (i) weakly stationary and (ii) invertible.
Define what is meant by an ARIMA (p, d, q) process. Let {Y} be such an ARIMA(p, d, q)
process and show how it can also be represented as an ARMA process, giving the
AR and MA orders of this representation.
(b) The following tables show the first nine sample autocorrelations and partial auto-
correlations of X and Y₁ = VX+ for a series of n = 1095 observations. (Notice
that the notation in this part has no relationship with the notation in part (a) of
this question.)
Identify a model for this time series and obtain preliminary estimates for the pa-
rameters of your model.
X₁
= 15.51, s² = 317.43.
k
1
2
3
4
5
6
7
Pk
0.981
0.974
0.968
akk 0.981 0.327
8
9
0.927
0.963 0.957 0.951 0.943 0.935
0.121 0.104 0.000 0.014 -0.067 -0.068 -0.012
Y₁ = VX : y = 0.03, s² = 11.48.
k
1…
Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.
Module Code: MATH380202
1. (a) Define the terms "strongly stationary" and "weakly stationary".
Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is
weakly stationary, define the autocorrelation function (acf) Pk, for lag k.
What conditions must a process {X+) satisfy for it to be white noise?
(b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of
the following processes {X+} are weakly stationary for t> 0? Briefly justify your
answers.
i. Xt for all > 0.
ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0.
(c) Provide an expression for estimating the autocovariance function for a sample
X1,..., X believed to be from a weakly stationary process. How is the autocor-
relation function Pk then estimated, and a correlogram (or acf plot) constructed?
(d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where
{E} is a white noise process with variance 1. Compute the population autocorre-
lation function Pk for all k = 0, 1, ....
Chapter 8 Solutions
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
Ch. 8.A - A town’s population increases in one year from...Ch. 8.A - A town’s population increases in one year from...Ch. 8.A - The balance owed your credit card doubles from...Ch. 8.A - The number Of songs in your iPod has increased...Ch. 8.A - Which of the following is in example of...Ch. 8.A - On a chessboard with 64 squares, you place 1 penny...Ch. 8.A - At 11:00 you place a single bacterium in a bottle,...Ch. 8.A - Consider the bacterial population described in...Ch. 8.A - Consider the bacterial population described in...Ch. 8.A - Which of the following is not true of any...
Ch. 8.A - Describe basic differences between linear growth...Ch. 8.A - 2. Briefly explain how repeated doublings...Ch. 8.A - Briefly summarize the Story Of the bacteria in the...Ch. 8.A - Explain the meaning Of the two key facts about...Ch. 8.A - Decide whether each of the following statements...Ch. 8.A - Suppose you had a magic hank account in which your...Ch. 8.A - A small town that grows exponentially can become a...Ch. 8.A - H. Human population has been growing exponentially...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Prob. 10ECh. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Prob. 12ECh. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - Linear or Exponential? State whether the growth...Ch. 8.A - 9-16: Linear or Exponential? State whether the...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Chessboard Parable. Use the chessboard parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Magic Penny Parable. Use the magic penny parable...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - Bacteria in a Bottle Parable. Use the bacteria...Ch. 8.A - 29. Human Doubling. Human population in the year...Ch. 8.A - Doubling Time versus Initial Amount. a. Would you...Ch. 8.A - Facebook Users. The table shows the number of...Ch. 8.A - Prob. 32ECh. 8.A - Exponential Growth. Identify at least two news...Ch. 8.A - Prob. 34ECh. 8.A - Prob. 35ECh. 8.B - Prob. 1QQCh. 8.B - Suppose your salary increases at a rate of 4.5%...Ch. 8.B - Which of the following is not a good approximation...Ch. 8.B - A town’s population doubles in 23 years. Its...Ch. 8.B - Radioactive tritium (hvdrogen-3) has a halt-life...Ch. 8.B - Radioactive uramum-235 has a hall-life of about...Ch. 8.B - The population of an endangered species decreases...Ch. 8.B - log10108= a.100,000,000 b. 108 c.8Ch. 8.B - A rural popular ion decreases at a rate of 20% per...Ch. 8.B - Prob. 10QQCh. 8.B - What is a doubling tune? Suppose a population has...Ch. 8.B - Prob. 2ECh. 8.B - State the approximate doubting time formula and...Ch. 8.B - Prob. 4ECh. 8.B - Prob. 5ECh. 8.B - 6. State the approximate hall-life formula and the...Ch. 8.B - 7. Briefly describe exact doubling time and...Ch. 8.B - 8. Give an example in which it is important to use...Ch. 8.B - Our town is growing with a doubling time of 25...Ch. 8.B - Our town is growing at a rate of 7% per year, so...Ch. 8.B - A toxic chemical decays with a hall-life of 10...Ch. 8.B - The hall-life of plutomum-239 is about 24,000...Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Prob. 15ECh. 8.B - Prob. 16ECh. 8.B - 13-24: Logarithms. Refer to the Brief Review on p....Ch. 8.B - Prob. 18ECh. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - Logarithms. Refer to the Brief Review on p. 488....Ch. 8.B - 13-24: Logarithms. Refer to the Brief Review on p....Ch. 8.B - Prob. 22ECh. 8.B - Prob. 23ECh. 8.B - 23-24. Logarithm Calculations
24. Using the...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - 25-32: Doubling Time. Each exercise gives a...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - Doubling Time. Each exercise gives a doubling time...Ch. 8.B - 25-32: Doubling Time. Each exercise gives a...Ch. 8.B - World Population. In 2017, estimated world...Ch. 8.B - World Population. In 2017, estimated world...Ch. 8.B - 31. Rabbits. A community of rabbits begins with an...Ch. 8.B - Prob. 36ECh. 8.B - Approximate Doubling Time Formula. Use the...Ch. 8.B - 37-40: Approximate Doubling Time Formula. Use the...Ch. 8.B - Approximate Doubling Time Formula. Use the...Ch. 8.B - 37-40: Approximate Doubling Time Formula. Use the...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - Half-Life. Each exercise gives a half-life for an...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41-48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41-48: Half-Life. Each exercise gives a half-life...Ch. 8.B - 41 -48: Half-Life. Each exercise gives a half-life...Ch. 8.B - Approximate Half-Life Formula. Use the approximate...Ch. 8.B - 49-52: Half-Life Formula. Use the approximate...Ch. 8.B - Approximate Half-Life Formula. Use the approximate...Ch. 8.B - 49-52: Half-Life Formula. Use the approximate...Ch. 8.B - Prob. 53ECh. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Exact Formulas. Compare the doubling times found...Ch. 8.B - Prob. 57ECh. 8.B - 58. Nuclear Weapons. Thermonuclear weapons use...Ch. 8.B - Prob. 59ECh. 8.B - Yucca Mountain. The U.S. government spent nearly...Ch. 8.B - Crime Rate. The homicide rate decreases at a rate...Ch. 8.B - Drug Metabolism. A particular antibiotic is...Ch. 8.B - Atmospheric Pressure. The pressure of Earth's...Ch. 8.B - Prob. 64ECh. 8.B - 65. Radioactive Half-Life. Find a news story that...Ch. 8.B - Prob. 66ECh. 8.B - Prob. 67ECh. 8.B - Prob. 68ECh. 8.B - Prob. 69ECh. 8.C - World population is currently rising by about 80...Ch. 8.C - The most recent doubling of world population took...Ch. 8.C - The primary reason for the rapid growth of human...Ch. 8.C - The carrying capacity of the Earth is defined as...Ch. 8.C - Which of the billowing would cause estimates of...Ch. 8.C - 6. Recall the bacteria in a bottle example from...Ch. 8.C - When researchers project that human population...Ch. 8.C - Prob. 8QQCh. 8.C - Prob. 9QQCh. 8.C - Prob. 10QQCh. 8.C - Based on Figure 8.3, contrast the changes in human...Ch. 8.C - Prob. 2ECh. 8.C - Haw do today’s birth and death rates compare to...Ch. 8.C - Prob. 4ECh. 8.C - Prob. 5ECh. 8.C - What is overshot and collapse? Under what...Ch. 8.C - Prob. 7ECh. 8.C - 8. If birth rates fall more than death rates, the...Ch. 8.C - The carrying capacity of our planet depends only...Ch. 8.C - to rapid increases in computing technology, we...Ch. 8.C - In the wild, we always expect the population of...Ch. 8.C - 12. Past civilizations that no longer exist must...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Varying Growth Rates. Starting from a 2017 world...Ch. 8.C - Birth and Death Rates. The following table gives...Ch. 8.C - 17-20: Birth and Death Rates. The following table...Ch. 8.C - 17-20: Birth and Death Rates. The following table...Ch. 8.C - Birth and Death Rates. The following table gives...Ch. 8.C - 21. Logistic Growth. Consider a population that...Ch. 8.C - Logistic Growth. Consider a population that begins...Ch. 8.C - U.S Population. Starting from an estimated U.S....Ch. 8.C - Prob. 24ECh. 8.C - Prob. 25ECh. 8.C - Prob. 26ECh. 8.C - Prob. 27ECh. 8.C - 28. Slower Growth. Repeat Exercise 27, but use a...Ch. 8.C - Prob. 29ECh. 8.C - World Carrying Capacity. For the given carrying...Ch. 8.C - 29-32: World Carrying Capacity. For the given...Ch. 8.C - 29-32: World Carrying Capacity. For the given...Ch. 8.C - Prob. 33ECh. 8.C - Prob. 34Ch. 8.C - Population Predictions. Find population...Ch. 8.C - Prob. 36ECh. 8.C - Prob. 37ECh. 8.C - Prob. 38ECh. 8.C - Prob. 39ECh. 8.C - Prob. 40ECh. 8.D - Prob. 1QQCh. 8.D - Prob. 2QQCh. 8.D - 3. What is a 0-decibel sound?
the softest sound...Ch. 8.D - 4. A sound of 85 decibels is defined to be
a. 85...Ch. 8.D - Prob. 5QQCh. 8.D - Prob. 6QQCh. 8.D - Prob. 7QQCh. 8.D - Prob. 8QQCh. 8.D - Prob. 9QQCh. 8.D - Prob. 10QQCh. 8.D - What is the magnitude scale for earthquakes? What...Ch. 8.D - What is the decibel scale? Describe how it is...Ch. 8.D - What is pH? What pH values define an acid, a base,...Ch. 8.D - What is acid rain? Why is it a serious...Ch. 8.D - An earthquake of magnitude 6 will do twice as much...Ch. 8.D - A 120-dB wand is 20% louder than a 100-dB sound.Ch. 8.D - If I double the amount of water in the cup, I'll...Ch. 8.D - The lake water was crystal clear, so It could not...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - Earthquake Magnitudes. Use the earthquake...Ch. 8.D - 9-14: Earthquake Magnitudes. Use the earthquake...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - 15-20: The Decibel Scale. Use the decibel scale to...Ch. 8.D - 15-20: The Decibel Scale. Use the decibel scale to...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - The Decibel Scale. Use the decibel scale to answer...Ch. 8.D - Prob. 20ECh. 8.D - Prob. 21ECh. 8.D - Prob. 22ECh. 8.D - Prob. 23ECh. 8.D - Inverse Square Law. Use the inverse square law for...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - 25-32: The pH Scale. Use the pH scale to answer...Ch. 8.D - Prob. 28ECh. 8.D - Prob. 29ECh. 8.D - Prob. 30ECh. 8.D - Prob. 31ECh. 8.D - Prob. 32ECh. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - 33-38: Logarithmic Thinking. Briefly describe, in...Ch. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - Logarithmic Thinking. Briefly describe, in words,...Ch. 8.D - Prob. 37ECh. 8.D - Prob. 38ECh. 8.D - 39. Sound and Distance.
The decibel level for...Ch. 8.D - 40. Variation in Sound with Distance. Suppose that...Ch. 8.D - Toxic Dumping in Acidified Lakes. Consider a...Ch. 8.D - Prob. 42ECh. 8.D - Earthquakes in the News. Find a recent news story...Ch. 8.D - Disasters. Find the death lolls for some of the...Ch. 8.D - Prob. 45ECh. 8.D - Prob. 46ECh. 8.D - Ocean Acidification. Research ocean acidification...
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