The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 8.2, Problem 14MS
The top 10 (ExH). Suppose you have 10 marbles. They are marked with the numbers 1, 2, 3, 4,5, 6, 7,8, 9, or 10. They are placed in a jar, and you reach in and select one. What is the probability that the number you select has a factor of 3? What is the probability that the number you select is a prime number? What is the probability that the number you select is even? What is the probability that the number you select is evenly divisible by 13?
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Problem 11 (a) A tank is discharging water through an orifice at a depth of T
meter below the surface of the water whose area is A m². The
following are the values of a for the corresponding values of A:
A 1.257 1.390
x 1.50 1.65
1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650
1.80 1.95 2.10 2.25 2.40 2.55 2.70
2.85
Using the formula
-3.0
(0.018)T =
dx.
calculate T, the time in seconds for the level of the water to drop
from 3.0 m to 1.5 m above the orifice.
(b) The velocity of a train which starts from rest is given by the fol-
lowing table, the time being reckoned in minutes from the start
and the speed in km/hour:
| † (minutes) |2|4 6 8 10 12
14 16 18 20
v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0
Estimate approximately the total distance ran in 20 minutes.
-
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
p-1
2
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
23
32
how come?
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
The set T is the subset of these residues exceeding
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Chapter 8 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 8.1 - Doors galore. The 21st-century version of Lets...Ch. 8.1 - Birthday surprise. How many people would you need...Ch. 8.1 - Opposite of heads. Suppose you flip a coin 100...Ch. 8.1 - Penny percent. Suppose you flip a penny 50 times,...Ch. 8.1 - Party time. At a nephews party, you decide to...Ch. 8.1 - Flipping Lincoln. Flip a penny 100 times and...Ch. 8.1 - Flashing cards. Shuffle a standard deck of 52...Ch. 8.1 - King for a day. Remove three cards from a deck of...Ch. 8.1 - A card deal stick. Remove three cards from a deck...Ch. 8.1 - A card deal switch. Remove three cards from a deck...
Ch. 8.1 - A card reunion—black first (S). Using a shuffled...Ch. 8.1 - A card reunion (H). Using a shuffled deck of...Ch. 8.1 - Prob. 14MSCh. 8.1 - Prob. 15MSCh. 8.1 - Personal perspectives. Write a short essay...Ch. 8.1 - With a group of folks. In a small group, discuss...Ch. 8.1 - One Grecian urn. A large urn in your kitchen is...Ch. 8.1 - Two Grecian urns. Your math instructor keeps two...Ch. 8.1 - Pennies from heaven (H). Residents of your...Ch. 8.1 - Dating Penny. The pennies described in the...Ch. 8.1 - Changing your pants. You have 20 coins in your...Ch. 8.2 - Black or white? Your friend chooses his sartorial...Ch. 8.2 - Eleven cents. You have a dime and a penny. Flip...Ch. 8.2 - Yummm. You have a small bag of candy-coated...Ch. 8.2 - Rubber duckies. A game at a carnival has 75 rubber...Ch. 8.2 - Legally large. What does the Law of Large Numbers...Ch. 8.2 - Lincoln takes a hit. On your wall is a poster...Ch. 8.2 - Giving orders. Order the following events in tenns...Ch. 8.2 - Two heads are better. Simultaneously flip a dime...Ch. 8.2 - Tacky probabilities. Before doing the following...Ch. 8.2 - BURGER AND STARBIRD. Suppose you randomly select a...Ch. 8.2 - Monty Hall. Read and rework the Lets Make a Deal...Ch. 8.2 - 7 or 11 (S). What is the probability of rollign a...Ch. 8.2 - D and D. You simultaneously flip a dime and roll a...Ch. 8.2 - The top 10 (ExH). Suppose you have 10 marbles....Ch. 8.2 - One five and dime (II). Someone simultaneously...Ch. 8.2 - Five flip. Someone flips five coins, but you dont...Ch. 8.2 - Flipped out. We take a coin and flip it 10,000,000...Ch. 8.2 - Spinning wheel. A roulette wheel has 36 spaces...Ch. 8.2 - December 9. Choose two people at random. What is...Ch. 8.2 - High roller (H). Using two fair dice, what is the...Ch. 8.2 - Double dice. You roll two fair dice. What is the...Ch. 8.2 - Sifly puzzle. After a professor explains the...Ch. 8.2 - Just do it. Find groups of roughly 35 people...Ch. 8.2 - No matches (S). Suppose 40 people are in a room....Ch. 8.2 - Spinner winner. If you were to spin the wheel...Ch. 8.2 - Flip side (S). Someone flips three coins behind a...Ch. 8.2 - Other flip side. Someone flips three coins behind...Ch. 8.2 - Blackjack. From a regular deck of 52 playing...Ch. 8.2 - Be rational (ExH). Suppose someone has randomly...Ch. 8.2 - Well red (H). Someone shows you three cards. One...Ch. 8.2 - Regular dice. Dungeons and Dragons players use...Ch. 8.2 - Take your seat. You decide to fly to California on...Ch. 8.2 - Eight flips. What is the probability of flipping a...Ch. 8.2 - Lottery (S). The lottery in an extremely small...Ch. 8.2 - Making the grade. What is wrong with the following...Ch. 8.2 - Dont squeeze. Five shoppers buy Charmin toilet...Ch. 8.2 - Birthday cards. Using a regular deck of 52 playing...Ch. 8.2 - Too many boys. Long, long ago and far, far away,...Ch. 8.2 - Three paradox (H). The correct probability of...Ch. 8.2 - Whats the big event? The section defines the...Ch. 8.2 - Tallying totals. For each given probability below,...Ch. 8.2 - Algebra roulette. Your math instructor has created...Ch. 8.2 - Probability beans. The jellybean jar on your math...Ch. 8.2 - Picking up the tab (H). Ed and Mike go out to...Ch. 8.3 - Daily deaths. About 58 million people die every...Ch. 8.3 - Wake-up call. Suppose you wake up each morning and...Ch. 8.3 - More than 12 monkeys. What does the Infinite...Ch. 8.3 - Get shorty. Who was Georges Louis Leclerc Comte de...Ch. 8.3 - Nothing but heads. If you flip a fair coin three...Ch. 8.3 - Pick a number. Pick a number from the following...Ch. 8.3 - Personal coincidences. List three coincidences you...Ch. 8.3 - No way. It is the last Sunday of spring break and...Ch. 8.3 - Prob. 10MSCh. 8.3 - Unlucky numbers. Suppose you randomly picked 1000...Ch. 8.3 - A bad block (S). Suppose 1054 people died in...Ch. 8.3 - Coin toss test. Ask two friends to help you repeat...Ch. 8.3 - DeceptIve dice (ExH). You asked three friends to...Ch. 8.3 - Murphys Law. If something can go wrong, it will....Ch. 8.3 - Drop the needle. Try Buffons needle experiment by...Ch. 8.3 - IBM again (H). Suppose we try the IBM...Ch. 8.3 - The dart index. Take a page of stock quotes from a...Ch. 8.3 - Random walks. Using a piece of graph paper, a...Ch. 8.3 - Random guesses (S). A multiple-choice test has 100...Ch. 8.3 - Random dates. There is a room filled with exotic...Ch. 8.3 - Random phones (H). Suppose you roll a 10-sided die...Ch. 8.3 - Fourever (ExH). Suppose you roll a die repeatedly,...Ch. 8.3 - Pick a number, revisited. In Mindscape 6, did you...Ch. 8.3 - Good start (H). Suppose the monkey is typing using...Ch. 8.3 - Even moves. Suppose you embark on a random walk on...Ch. 8.3 - Playing the numbers. Here is a numbers game. You...Ch. 8.3 - Random results. Someone looks at a list of 10,000...Ch. 8.3 - Monkey names. Suppose we have a monkey typing on a...Ch. 8.3 - The streak. Suppose you flip a fair coin 10 times...Ch. 8.3 - Girl, Girl,... (S). A couple has eight children....Ch. 8.3 - One mistake is okay. Suppose we try the IBM...Ch. 8.3 - Picking and matching. You and a friend...Ch. 8.3 - Picking and matching. You and a friend...Ch. 8.3 - Dice are different (ExH). In Mindscape 14 you...Ch. 8.3 - Death row (H). You may have noticed that two pairs...Ch. 8.3 - Striking again. Consider the striking deal game...Ch. 8.3 - Random returns. Suppose we take a random walk on...Ch. 8.3 - Random natural. Suppose you have a 10-sided die...Ch. 8.3 - Ace of spades. You randomly shuffle a deck of...Ch. 8.3 - Flipping out. You flip a coin eight times and get...Ch. 8.3 - Random reigns (H). Your (very egalitarian) school...Ch. 8.3 - Survey says... A survey in your intro psychology...Ch. 8.3 - Rock n roll. As a project in your Geology and Art...Ch. 8.3 - Random pitch. During a practice game, your...Ch. 8.4 - Have a heart. If you draw a card from a regular...Ch. 8.4 - Have a head. You ffip a coin 10 times. What is the...Ch. 8.4 - Sunny surprise. Suppose the chances are 1 in 2...Ch. 8.4 - Elephant ears. Suppose a quarter of all the...Ch. 8.4 - Little deal. In how many ways can you select three...Ch. 8.4 - The gym lock. A lock has a disk with 36 numbers...Ch. 8.4 - The dorm door. A dormitory has an electronic lock....Ch. 8.4 - 28 cents. How many different ways can you make 28...Ch. 8.4 - 82 cents. How many different ways can you make 82...Ch. 8.4 - Number please. Someone you really wanted to go out...Ch. 8.4 - Dealing with jack. Suppose you deal three cards...Ch. 8.4 - MA Lotto (H). To win the jackpot of the...Ch. 8.4 - NY Lotto (H). To win the jackpot of the New York...Ch. 8.4 - OR Lotto. To win the jackpot of the Oregon lottery...Ch. 8.4 - Burger King (S). You take a summer job making...Ch. 8.4 - More burgers. Suppose you are working at a burger...Ch. 8.4 - NetFlix. You have 65 movies on your NetFlix queue....Ch. 8.4 - Cineplex (ExH). Your local Cineplex is showing...Ch. 8.4 - One die. You roll a fair die four times. What is...Ch. 8.4 - Dressing for success. You have 5 T-shirts, 10...Ch. 8.4 - Band stand. The Drew Aderburg Band is planning a...Ch. 8.4 - Mondays undies. You are spending the weekend at a...Ch. 8.4 - Prob. 23MSCh. 8.4 - Cranking tunes. Your car stereo can be programmed...Ch. 8.4 - The Great Books. There are 20 Great Books, from...Ch. 8.4 - Morning variety (S). You wish to have a different...Ch. 8.4 - Blind mans bluff. You and a friend each pick a...Ch. 8.4 - Crime story (ExH). Suppose 20 witnesses saw...Ch. 8.4 - Theres a 4 (H). Someone you really want to go out...Ch. 8.4 - Making up the test. Your math prof says there will...Ch. 8.4 - Moving up. You have a part4ime job in a department...Ch. 8.4 - Counterfeit bills. You are given ten $100 bills...Ch. 8.4 - Car care. A burglar wishes to break into a car...Ch. 8.4 - Coins count. On your bureau you have a half...Ch. 8.4 - Math mania. There are 90 students enrolled in Math...Ch. 8.4 - Party on. You want to throw a party and can invite...Ch. 8.4 - No dice (H). You roll a pair of dice 24 times....Ch. 8.4 - Three angles. Draw 10 points on a piece of paper...Ch. 8.4 - Four parties (ExH). You want to have a party and...Ch. 8.4 - Making the cut. In 1988, the ignition keys for...Ch. 8.4 - Parental pride. The number of pictures you have of...Ch. 8.4 - Sister, Sister. The number of students in a...Ch. 8.4 - In search of a cubic. This section explains that...Ch. 8.4 - Cancellation! (H) This section explains that the...Ch. 8.4 - Keep on cancellin! Building on the previous...Ch. 8.5 - No pop quizzes. Your instructor gives mom quizzes...Ch. 8.5 - No easy quizzes (S). Your instructor starts giving...Ch. 8.5 - Silly sickness. Based on your symptoms, your...Ch. 8.5 - Making an algebra-down. Consider the equation...Ch. 8.5 - A beautiful mind. Who was John Nash? Give as many...Ch. 8.5 - Upon further study. On your instructor quizzes,...Ch. 8.5 - Power pass (ExH). The matrix below gives the yards...Ch. 8.5 - Pass with a p. Suppose the defense referred to in...Ch. 8.5 - Random run. Suppose now in Mindscape 7 the defense...Ch. 8.5 - Another random nm. Suppose now the defense...Ch. 8.5 - Lining up the defense (S). Using the payoff matrix...Ch. 8.5 - De-lines of defense. Using the payoff matrix in...Ch. 8.5 - Approximate Nash (ExH). Take the two lines from...Ch. 8.5 - Precise Nash. Use your answers to Mindscapes 8 and...Ch. 8.5 - Positive payoof (H). Below is a payoff matrix for...Ch. 8.5 - Estimating the equilibrium. Estimate the...Ch. 8.5 - Exacting the equilibrium. Find equations for the...Ch. 8.5 - Payoffs need not be positive. Heres a payoff...Ch. 8.5 - Negative equilibrium. Find equations for the two...Ch. 8.5 - Making negative sense. In what kind of game or...Ch. 8.5 - Colin and Tubbes. When first grader Cohn comes...Ch. 8.5 - Cohn and Tubbes 50—50. Suppose in Mindscape 21...Ch. 8.5 - Averaging Cohn and Tubbes. Suppose p is the...Ch. 8.5 - Colin and Tubbes line up (ExH). Referring to...Ch. 8.5 - Colin and Tubbes with a dash of Nash (S). Find the...Ch. 8.5 - Diagnosis. Based on your symptoms, your doctor...Ch. 8.5 - Sploosh test (H). Given the scenario in the...Ch. 8.5 - Diagnose the data. Using the data from the...Ch. 8.5 - My theory ... You are a paleontologist who finds...Ch. 8.5 - Going stag. Two Englishmen, Neville and Winston,...Ch. 8.5 - Selling sweets. Two snack cake companies, Snacky...Ch. 8.5 - Prisoners dilemma. Two people are suspected of...Ch. 8.5 - Match point. Find the coordinates (p,y) where the...Ch. 8.5 - Match lines. Find an equation for the line passing...Ch. 8.5 - Unmatched slopes. Two lines intersect at the point...Ch. 8.5 - Plotting percentages. Of the students in your...Ch. 8.5 - Itching for x (H). You go to the Health Center...
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