Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Mathematical Excursions (MindTap Course List)
EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.EXCURSION EXERCISES Solve each of the, following puzzles. Note: The authors of this textbook are not associated with the KenKen brand. Thus the following puzzles are not official KenKen puzzles; however, each puzzle can be solved using the same techniques one would use to solve an Official KenKen puzzle.Use inductive reasoning to predict the next number in each list. 4, 8, 12, 16, 20, 24 ,?Use inductive reasoning to predict the next number in each list. 5, 11, 17, 23, 29, 35 ,?Use inductive reasoning to predict the next number in each list. 3, 5, 9, 15, 23, 33 ,?Use inductive reasoning to predict the next number in each list. 1, 8, 27, 64, 125 ,?Use inductive reasoning to predict the next number in each list. 1, 4, 9, 16, 25, 36, 49 ,?Use inductive reasoning to predict the next number in each list. 80, 70, 61, 53, 46, 40 ,?Use inductive reasoning to predict the next number in each list. 35, 57, 79, 911, 1113, 1315 ,?Use inductive reasoning to predict the next number in each list. 12, 23, 34, 45, 56, 67 ,?Use inductive reasoning to predict the next number in each list. 2, 7, 3, 2, 8, 3, 13, 8, 18 ,?Use inductive reasoning to predict the next number in each list. 1, 5, 12, 22, 35 ,?Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… The sum of any two even counting numbers is always an even counting number.Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… The product of an odd counting number and an even counting number is always an even counting number.Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… The product of two odd counting numbers is always an odd counting number.Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… The sum of two odd counting numbers is always an odd counting number.Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… Pick any counting number. Multiply the number by 6. Add 8 to the product. Divide the sum by 2. Subtract 4 from the quotient. The resulting number is twice the original number.Use inductive reasoning to decide whether each statement is correct. Note: The numbers 1, 2, 3, 4, 5, are called taunting numbers or natural numbers. Any counting number n divided by 2 produces a remainder of 0. If n2 has a remainder of 0, then n is an even counting number. If n2 has a remainder of 1, then is an odd counting number . Even counting numbers: 2, 4, 6, 8, 10,… Odd counting numbers: 1, 3, 5, 7, 9,… Pick any counting number. Multiply the number by 8. Subtract 4 from the product. Divide the difference by 2. Add 2 to the quotient. The resulting number is four times the original number.Determine the distance a ball rolls, on inclined plane 1, during each of the following time intervals. Hint: To determine the distance a ball rolls in the interval t=1 to t=2 seconds, find the distance it rolls in 2 seconds and from this distance subtract the distance it rolls in 1 second. a. 1st second: t=0 to t=1 second b. 2nd second: t=1 to t=2 seconds c. 3rd second: t=2 to t=3 seconds d. 4th second: t=3 to t=4 seconds e. 5th second: t=4 to t=5 secondsDetermine the distance a ball rolls, on inclined plane2, during each of the following time intervals. a. 1st second:t=0 tot=1 second b. 2nd second:t=1 tot=2 seconds c. 3rd second:t=2 tot=3 seconds d. 4th second:t=3 tot=4 seconds e. 5th second:t=4 tot=5 secondsFor inclined plane 1, the distance a ball rolls in the 1st second is 8 centimeters. Think of this distance as 1 unit. That is, for inclined plane 1, 1 unit = 8 centimeters. Determine how farin terms of units a ball will roll, on inclined plane 1, in the following time intervals. a. 2nd second: t=1 to t=2 seconds b. 3rd second: t=2 to t=3 seconds c. 4th second: t=3 to t=4 seconds d. 5th second: t=4 to t=5 secondsFor inclined plane 2, the distance a ball rolls in the 1st second is 6.5 centimeters. Think of this distance as 1 unit. Determine how far in terms of units a ball will roll, on inclined plane 2, in the following time intervals. a. 2nd second: t=1 to t=2 seconds b. 3rd second: t=2 to t=3 seconds c. 4th second: t=3 to t=4 seconds d. 5th second: t=4 to t=5 secondsUse inductive reasoning and the data in the inclined plane time-distance table, shown above exercise 17 predict the answer to each question. If the time a ball is allowed to roll on an inclined plane is doubled, what effect does this have on the distance the ball rolls?Use inductive reasoning and the data in the inclined plane time-distance table, shown above exercise 17 predict the answer to each question. If the time a ball is allowed to roll on an inclined plane is tripled, what effect does this have on the distance the ball rolls?Use inductive reasoning and the data in the inclined plane time-distance table, shown above exercise 17 predict the answer to each question. How far will a hall roll an inclined plane 1 in 6 seconds?Use inductive reasoning and the data in the inclined plane time-distance table, shown above exercise 17 predict the answer to each question. How far will a ball roll an inclined plane 1 in 1.5 seconds?Determine whether the argument is an example of inductive reasoning or deductive reasoning. Emma enjoyed reading the novel Finders Keepers by Stephen King, 50 she will enjoy reading his next novel.Determine whether the argument is an example of inductive reasoning or deductive reasoning. All pentagons have exactly five sides. Figure A is a pentagon. Therefore, Figure A has exactly five sides.Determine whether the argument is an example of inductive reasoning or deductive reasoning. Every English setter likes to hunt. Duke is an English setter, 50 Duke likes to hunt.Determine whether the argument is an example of inductive reasoning or deductive reasoning. Cats dont eat tomatoes. Tigger is a cat. Therefore, Tigger does not eat tomatoes.Determine whether the argument is an example of inductive reasoning or deductive reasoning. A number is neat number if the sum of the cubes of its digits equals the number. Therefore, 153 is number.Determine whether the argument is an example of inductive reasoning or deductive reasoning. The Atlanta Braves have won five games in a raw. Therefore, The Atlanta Braves will win their next game.Determine whether the argument is an example of inductive reasoning or deductive reasoning. Since 11(1)(101)=111111(2)(101)=222211(3)(101)=333311(4)(101)=444411(5)(101)=5555 we know that the product of 11 and a multiple of 101 is a number in which every digit is the same.Determine whether the argument is an example of inductive reasoning or deductive reasoning. The following equations show that n2n+11 is a prime number for all counting numbers is = 1, 2, 3 , 4 ,… (1)21+11=11 n=1 (2)22+11=13 n=2 (3)23+11=17 n=3 (4)24+11=23 n=4 Note: A prime number is a counting number greater than 1 that has no counting number factors other than itself and 1. The first 15 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, x1x.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, x+xx.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, x3x.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, | x+3 |=| x |+3.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, xx.Find a number that provides a counterexample to show that the given statement is false. For all numbers x, (x+1)(x1)(x1)=x+1.Find a pair of numbers that provides a counterexample to show that the given statement is false. If the sum of two counting numbers is an even counting number, then the product of the two counting numbers is an even counting number.Find a pair of numbers that provides a counterexample to show that the given statement is false. If the product of two counting numbers is an even counting number, then both of the counting numbers are even counting numbers.Use deductive reasoning to determine the missing numbers in each magic square. A magic square of order 4Use deductive reasoning to determine the missing numbers in each magic square. A magic square of order 5Use deductive reasoning to show that the following procedure always produces a number that is equal to the original number. Procedure: Pick a number. Multiply the number by 6 and add 8. Divide the sum by 2, subtract twice the original number, and 4.Use deductive reasoning to show that the following procedure always produces the number 5. Procedure: Pick a number. Add 4 to the number and multiply the sum by 3. Subtract 7 and then decrease this difference by the triple of the original number.Stocks Each of four siblings (Anita, Tony, Maria and Jose) is given 5000 to invest in the stock maker. Each chooses a different stock. One chooses a utility shock, another an automotive stock, another a technology shock, and the other an oil stock. From the following clues, determine which sibling bought which stock. a. Anita and the owner of the utility stock purchased their shares through an online brokerage, whereas Tony and the owner of the automotive stock did not. b. The gain in value of Maria's stock is twice the gain in value of the automotive stock. c. The technology stock is traded on NASDAQ, whereas the stock that Tony bought is traded on the New York Stock Exchange.Gourmet Chefs The Changs, Steinbergs, Ontkeans, and Gonzaleses were winners in the All-State Cooking Contest. There was a winner in each of four categories: soup, entrĂ©e, salad, and dessert. From the following clues, determine in which category each family was the winner. a. The soups were judged before the Ontkeans' winning entry. b. This year's contest was the first for the Steinbergs and for the winner in the dessert category. The Changs and the winner of the soup category entered last year's contest. c. The winning entrĂ©e took 2 hours to cook, whereas the Steinbergs entrĂ©e required no cooking at all.Collectibles The cities of Atlanta, Chicago, Philadelphia, and San Diego held conventions this summer for collectors of coins, stamps, comic books and base-ball cards. From the following clues, determine which collectors met in which city. a. The comic book collectors convention was in August, as was the convention held in Chicago. b. The baseball card collectors did not meet in Philadelphia, and the coin collectors did not meet in San Diego or Chicago. c. The convention in Atlanta was held during the week of July 4, whereas the coin collector convention was held the week after that. d. The convention in Chicago had more collectors attending it than did the stamp collectors convention.48ESDriving Time You need to buy groceries at the supermarket, deposit a check at the credit union, and purchase a book at the bookstore. You can complete the errands in any order; however, you must start and end at your home. The driving time in minutes between each of these locations is given in the following figure. Find a route for which total driving time is less than 30 minutes.Driving Time Suppose, that you need to go to the supermarket after you have completed the other two errands. What route should you lake to minimize you travel time?Use inductive reasoning to predict the next letter in the following list. O, T, T, F, F, S, S, E, ,,, Hint: Look for a pattern that involves letters from words used for counting.52ESCounterexamples Find a counterexample to prove that the inductive argument in a. Exercise 31 is incorrect. b. Exercise 32 is incorrect.54ESExtend Figure 1.1 above by constructing drawings of the sixth triangular number, the sixth square number, and the sixth pentagonal number.The figure below shows that the fourth triangular number, 10, added to the fifth triangular number, 15, produces the fifth square number, 25. a. Use a drawing to show that the fifth triangular number added to the sixth triangular number is the sixth square number. b. Verify that the 50th triangular number added to the 51st triangular number is the 51st square number. Hint: Use a numerical approach; dont use a drawing. c. Use nth-term formulas to verify that the sum of the nth triangular number and the (n+1)st triangular number is always the square number (n+1)2.Construct a drawing of the fourth hexagonal number.Construct a difference table to predict the next term of each sequence. 1,7,17,31,49,71,...Construct a difference table to predict the next term of each sequence. 10,10,12,16,22,30,...Construct a difference table to predict the next term of each sequence. 1,4,21,56,115,204,...Construct a difference table to predict the next term of each sequence. 0,10,24,56,112,190,...Construct a difference table to predict the next term of each sequence. 9,4,3,12,37,84,...Construct a difference table to predict the next term of each sequence. 17,15,25,53,105,187,...Use the given nth-term formula to compute the first five terms of the sequence. an=n(2n+1)2Use the given nth-term formula to compute the first five terms of the sequence. an=nn+1Use the given nth-term formula to compute the first five terms of the sequence. an=5n23nUse the given nth-term formula to compute the first five terms of the sequence. an=2n3n2Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figureCannonballs can be stacked to form a pyramid with a triangular base. Five of these pyramids are shown below. Use these figures. a. Use a difference table to predict the number of cannonballs in the sixth pyramid and in the seventh pyramid. b. Write a few sentences that describe the eighth pyramid in the sequence.Cannonballs can be stacked to form a pyramid with a triangular base. Five of these pyramids are shown below. Use these figures. The sequence formed by the numbers of cannonballs in the above pyramids is called the tetrahedral sequence. The nth-term formula for the tetrahedral sequence is Tetrahedraln=16n(n+1)(n+2). Find Tetrahedral10.Pieces vs. Cuts One cut of a stick of licorice produces two pieces. Two cuts produce three pieces. Three cuts produce four pieces. a. How many pieces are produced by five cuts and by six cuts? b. Predict the nth-term formula for the number of pieces of licorice that are produced by n cuts of a stick of a licorice.Pieces vs. Cuts One straight cut across a pizza produces 2 pieces. Two cuts can produce a maximum of 4 pieces. Three cuts can produce a maximum of 7 pieces. Four cuts can produce a minimum of 11 pieces. a. Use a difference table to predict the maximum number of pieces that can be produced with seven cuts. b. How are the pizza-slicing numbers related to the triangular numbers, which are defined by Triangularn=n(n+1)2?Pieces vs. Cuts One straight cut through a thick piece of cheese produces two pieces. Two straight cuts can produce a maximum of 4 places. Three straight cuts can produce a maximum 8 pieces. You might be inclined to think that every additional cut doubles the previous number of pieces. However, far four straight cuts, you will find that you get a maximum of 15 pieces. All nth-term formula for the maximum number of pieces, Pn, that can be produced by n straight cuts is Pn=n3+5n+66 a. Use the nth-term formula to determine the maximum number of pieces that can be produced by five straight cuts. b. What is the smallest number of straight cuts that you can use if you wish to produce at least 60 pieces? Hint: Use the nth-term formula and experiment will larger and larger values of n.Fibonacci Properties The Fibonacci sequence has many unusual properties. Experiment to decide which of the following properties are valid. Note: Fn represents the nth Fibonacci number. a. 3FnFn2=Fn+2 for n3 b. FnFn+3=Fn+1Fn+2 c. F3n is an even number. d. 5Fn2Fn2=Fn+3 for n3Find the third, fourth, and fifth terms of the sequence defined by a1=3, a2=5, and an=2an1an2 for n3.Find the third, fourth, and fifth terms of the sequence defined by a1=2, a2=3, and an=(1)an1+an2, for n3.Binets Formula The following formula is known as Binet's formula for the nth Fibonacci number. Fn=15[ (1+52)n(152)n ] The advantage of this formula over the recursive formula Fn=Fn1+Fn2 is that you can determine the nth Fibonacci number without finding the two pre- ceding Fibonacci numbers. Use Binets formula and a calculator find the 20th. 30th, and 40th Fibonacci numbers.Binets Formula Simplified Binets formula (see Exercise 23) can be simplified if you round your calculator results to the nearest integer. In the following Formula, nint is an abbreviation for the nearest integer of. Fn=nint{ 15(1+52)n } If you use n=8 in the above formula, a calculator will show 21.00951949 for the value inside the braces. Rounding this number to the nearest integer produces 21 as the eighth Fibonacci number. Use the above form of Binets formula and a calculator to find the 16th, 21st, and 32nd Fibonacci numbers.A Geometric Model The ancient Greeks often discovered mathematical relationships by using geometric drawings. Study the accompanying drawing to determine what need to put in place of the question mark to make the equation a true statement. 1+3+5+7+...+(2n1)=?The nth-term formula an=n(n1)(n2)(n3)(n4)4321+2n generates 2, 4, 6, 8, 15 for n=1,2,3,4,5. Make minor changes m to above formula to produce an nth-term formula (with n=l,2,3,4,and5 ) that will generate the following finite sequences. a. 2, 4, 6, 8, 20 b. 2, 4, 6, 8, 30Fibonacci Sums Make a conjecture for each of the following sums, where Fn represent the nth Fibonacci number. a. Fn+2Fn+1+Fn+2=? b. Fn+Fn+1+Fn+3=?Fibonacci Sums Make a conjecture for each of the following sums, where Fn represent the nth Fibonacci number. a. F1+F2+F3+F4+...+Fn=? b. F2+F4+F6+...+F2n=?Pascals Triangle The triangular pattern in the following figure is known as Pascals triangle. Pascal's triangle has intrigued mathematician for hundreds of years. Although it is named after the mathematician Blaise Pascal (1623-1662), there is evidence that it was first developed in China in the 1300s. The numbers in Pascal's triangle are created in the following manner. Each row begins and ends with the number 1. Any other number in a row is the sum of the two closest numbers about it. For instance, the first 10 in raw 5 is the sum of the first 4 and the 6 above it in raw 4. There are many patterns that can be discovered in Pascal's triangle. a. Find the sum of the numbers in each row, except row 0, of the portion of Pascal's triangle shown above. What pattern do you observe concerning mesa sums? Predict the sum of the numbers in row 9 of Pascals triangle. b. The numbers 1,3,6,10,15,...,n(n+1)2,... are called triangular number. Where do the triangular number appear in Pascals triangle?A Savings Plan You save a penny on day 1. On each of the following days you save double the amount of money you saved on the previous day. How much money will you have after: a. 5 days? b. 10 days? c. 15 days? d. n days? Hint: 21=2,22=4,23=8,24=16, 25=32,...,210=1024,...,215=32758,...A Famous Puzzle The Tower of Hanoi is a puzzle invented by Edouard Lucas in 1883. The puzzle consists of three pegs and a number of disks of distinct diameters stacked on one of the pegs such that the largest disk is on the bottom, the next largest is placed on the largest disk, and so on as shown on page 26. The object of the puzzle is to transfer the tower to one of the other pegs. The rules require that only one disk be moved at a time and that a larger disk may not be placed on a mailer disk. All pegs may be used. Determine the minimum number of moves required to transfer all of the disks to another peg for each of the following situations. a. You start with only one disk. b. You start with two disks. c. You start with three disks. (Note: You can use a stack of various size coins to simulate the puzzle, or you can use one of the many websites that provide a simulation of the puzzle.) d. You start with four disks. e. You start with five disks. f. You start with n disks. g. Lucas included with the Tower puzzle a legend about a tower that had 64 gold disks on one of three diamond needles. A group of priests had the task of transferring the 64 disks to one of the other needles using the same rules as the Tower of Hanoi puzzle. When they had completed the transfer, the tower would crumble and the universe would cease to exist. Assuming that the priests could transfer one disk to another needle every second, how many years would it take them to transfer all of the 64 disks to one of the other needles?Use the recursive definition for Fibonacci numbers and deductive reasoning to verify that, for Fibonacci numbers, 2FnFn2=Fn+1 for n3. Hint: By definition. Fn+1=Fn+Fn1 and Fn=Fn1+Fn2.Use the probability demonstrator, in the left margin, to work Excursion. How many routes can a ball take as it travels from A to B, from A to C, from A to D, from A to E, and from A to F? Hint: This problem is similar to Example 1 on page 27.Use the probability demonstrator, in the left margin, to work Excursion. How many routes can a ball take as it travels from A to G, from A to H, from A to L from A to J, and from A to K?Use the probability demonstrator, in the left margin, to work Excursion. Explain how you know that the number of routes from A to J is this same as the number of routes from A to L.Use the probability demonstrator, in the left margin, to work Excursion. Explain why the greatest number of balls tend to fall into the center bin.Use the probability demonstrator, in the left margin, to work Excursion. The probability demonstrator Shawn to the right has nine rows of hexagons. Determine how many routes a ball can lake as it travels from A to P, from A to Q, from A to R, from A to S, from A to T, and from A to U.Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Girls There are 364 first-grade students in Park Elementary School. If there are 26 more girls than boys, how many girls are there?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Heights of Ladders If two ladders are placed end to end, their combined height is 31.5 feet. One ladder is 6.5 feet shorter than the other ladder. What are the heights of the two ladders?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Squares How many squares are in the following figure?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Determine a Digit What is the 44th decimal digit in the decimal representation of 111? 111=0.09090909...Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Cost of a Shirt A shirt and a tie together cost $50. The shirt costs $30 more than the tie. What is the cost of the shirt?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Games In a basketball league consisting of 12 teams, each team plays each of the other learns exactly twice. How many league games will be played?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Routes Consider the following map. Tyler wishes to walk along the streets from point A to point B. How many direct routes (no backtracking] can Tyler take?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Routes Use the map in Exercise 7 to answer each of the following. a. How many direct routes are there from A to B if you want to pass by Starbucks? b. How many direct routes are there from A to B if you want to stop at Subway for a sandwich? c. How many direct routes are there from A to B if you want to stop at Starbucks and at Subway?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. True-False Test In how many ways can you answer a 12-question true-false test if you answer each question with either a true or a false?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. A Puzzle A frog is at the bottom of a 17-foot well. Each time the frog leaps, it moves up 3 feet. If the frog has not reached the top of the well, then the frog slides back 1 foot before it is ready to make another leap. How many leaps will the frog need to escape the well?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Handshake If eight people greet each other at a meeting by shaking hands with one another, how many handshakes take place?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Line Segments Twenty-four points are placed around a circle. A line segment is drawn between each pair of points. How many line segments are drawn?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Number of Ducks and Pig The number of ducks and pigs in a field totals 35. The total number of legs among them is 98. Assuming each dunk has exactly two legs and each pig has exactly four legs, determine how many duels and how many pigs are in the field.Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Racing Strategies Carla and Allison are sisters. They are on their way from school to home. Carla runs half the lime and walks half the time. Allison runs half the distance and walks half the distance. If Carla and Allison walk at the same speed and run at the same speed, which one arrives home first? Explain.Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Change for a Quarter How many ways can you make change for 25¢ using dimes, nickels, and/or pennies?Use Polyas four-step problem-solving strategy and the problem-solving procedures presented in this section to solve each of the following exercises. Carpet for a Room A room measures 12 feet by 15 feet. How many 3-foot by 3-foot square of carpet are needed to cover the floor of this room?Determine the Units Digit Determine the units digit (one digit) of the counting number represented by the exponential expression. 4300Determine the Units Digit Determine the units digit (one digit) of the counting number represented by the exponential expression. 2725Determine the Units Digit Determine the units digit (one digit) of the counting number represented by the exponential expression. 3412Determine the Units Digit Determine the units digit (one digit) of the counting number represented by the exponential expression. 7146Find Sums Find the following sums without using a calculator. Hint: Apply the procedure used by Gauss. (See the Math Matters on page 31.) a. 1+2+3+4+...+397+398+399+400 b. 1+2+3+4+...+547+548+549+550 c. 2+4+6+8+...+80+82+84+86Explain how you could modify the procedure used by Gauss (see the Math Matters on page 31) to find the following sum. 1+2+3+4+...+62+63+64+65Palindromic Numbers A palindromic number is a whole number that remains unchanged when its digits are written in reverse order. Find all palindromic numbers that have exactly a. three digits and are the square of a natural number. b. four digits and are the cube of a natural number.Speed of a Car A car has an odometer reading of 15,951 miles, which is a palindromic number. (See Exercise 23.) After 2 hours of continuous driving at a constant speed, the odometer reading is the next palindromic number. How fast, in miles per hour, was the car being driven during these 2 hours?A Puzzle Three volumes of the series Mathematics: Its Content, Methods, and Meaning are on a shelf with no space between the volumes. Each volume is 1 inch thick without its covers. Each cover is 18 inch thick. See the following figure. A bookworm bores horizontally from the first page of Volume 1 to the last page of Volume III. How far does the bookworm travel?Connect the Dots Nine dots are arranged as shown. Is it possible to connect the nine dots with exactly four lines if you are not allowed to retrace any part of a line and you are not allowed to ram your pencil from the paper? If it can be done, demonstrate with a drawing.Movie Theatre Admissions The following bar graph shows the number of U.S. and Canada movie theatre admissions for the years from 2007 to 2014. a. Estimate the number of admissions for the year 2009. Round to the nearest tenth of a billion. b. Which year had the least number of admissions? c. Which year had the greatest number of admissions?Box Office Revenues The following broken-line graph shows the U.S. and Canada movie theatre box office revenues, in billions, of dollars, for the years from 2007 to 2014. a. Which two years had the least box office revenues? b. Which years had the greatest box office revenue? c. During which two consecutive years did the box office revenues increase the most?Movie Ratings and Box Office Revenue The following circle graph shows the percentage of the 10.4 billion dollar box office revenue attributed to each of the various movie ratings in 2014. a. Which mantle rating brought in the largest share of the 2014 box office revenue? b. Determine the 2014 box office revenue produced by the PG-rated films. Round to the nearest tenth of a billion dollars.Votes in an Election In a school election one candidate for class president received more than 94%. But less than 100% of the votes cast. What is the least possible number of vanes cast?Floor Design A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are blue. The rest of the tiles are green. If 101 blue tiles are used, find the total number of tiles on the floor.Number of Children How many children are there in a family wherein each girl has as many brothers as sisters, but each boy has twice as many sisters as brothers?Brothers and Sisters I have two more sisters than brothers. Each of my sisters has two more sisters than brothers. How many more sisters than brothers does my youngest brother have?A Coin Problem If you take 22 pennies from a pile of 57 pennies, how many pennies do you have?Bacterial Growth The bacteria in a petri dish grow in a manner such that each day the number of bacteria doubles. On what day will the number of bacteria be half of the number present on the 12th day?Number of River Crossings Four people on one side of a river need to cross the river in a boat that can carry a maximum load of 180 pounds. The weights of the people are 80, 100, 150, and 170 pounds. a. Explain how the people can use the bout to gel everyone to the opposite side of the river. b. What is the minimum number of crossings that must be made by the boat?Examination Scores On three examinations, Dana received scores of 82, 91, and 76. What score does Dana need m the fourth examination to raise his average to 85?Puzzle from a Movie In the movie Die Hard: With a Vengeance. Bruce Willis and Samuel L. Jackson are given a 5-gallon jug and a 3-gallon jug and they must put exactly 4 gallon of water on a scale to keep a bomb from exploding. Explain how they could accomplish this feat.Find the Fake Coin You have eight coins. They all lock identical, but one is a fake and is slightly lighter than the others. Explain how you can use a balance scale to determine which coin is the fake in exactly a. three weighings. b. two weighings.Problems from the Mensa Workout Mensa is a society that welcomes people from every walk of life whose IQ is in the top 2% of the population. The multiple-choice Exercises 40 to 43 are from the Mensa Workout, which is posted on the Internet at www.mensa.org If it were 2 hours later, it would be half as long until midnight as it would be if it were an hour later. What time is it now? a. 18:30 b. 20:00 c. 21:00 d. 22:00 e. 23:30Problems from the Mensa Workout Mensa is a society that welcomes people from every walk of life whose IQ is in the top 2% of the population. The multiple-choice Exercises 40 to 43 are from the Mensa Workout, which is posted on the Internet at www.mensa.org Sally likes 225 but not 224; she likes 900 but not 800; she likes 144 but not 145. Which of the following does she like? a. 1600 b. 1700Problems from the Mensa Workout Mensa is a society that welcomes people from every walk of life whose IQ is in the top 2% of the population. The multiple-choice Exercises 40 to 43 are from the Mensa Workout, which is posted on the Internet at www.mensa.org There are 1200 elephants in a herd. Some have pink and green stripes, some are all pink, and some are all blue. One third are pure pink. Is it true that 400 elephants are definitely blue? a. Yes b. NoProblems from the Mensa Workout Mensa is a society that welcomes people from every walk of life whose IQ is in the top 2% of the population. The multiple-choice Exercises 40 to 43 are from the Mensa Workout, which is posted on the Internet at www.mensa.org Following the pattern shown in the number sequence below, what is the missing number? 1 8 27 ? 125 216 a. 36 b. 45 c. 46 d. 64 e. 99Compare Exponential Expressions a. How many times as large is 3(33) than (33)3? b. How many times as large is 4(44) than (44)4? Note: Most calculators will not display the answer to this problem because it is too large. However, the answer can be determined in exponential form by applying the following properties of exponents. (am)n=amnandaman=amnA Famous Puzzle The mathematician Augustus De Morgan once wrote that he had the distinction of being x years old in the year x2. He was 43 in the year 1849. a. Explain why people born in the year 1980 might share the distinction of being x years old in the year x2. Note: Assume x is a natural number. b. What is the next year after 1980 for which people born in that year might be x years old in the year x2?Verify a Procedure Select a two-digit number between 50 and 100. Add 83 to your number. From This number form a new number by adding the digit in the hundreds place to the number formed by the other two digits (the digits in the tens place and the ones place). Now subtract this newly formed number from your original number. Your final result is 16. Use a deductive approach to show that the final result is always 16 regardless of which number you start with.Numbering Pages How many digits does it take in total to number a book from page 1 to page 240?Mini Sudoku Sudoku is deductive reasoning number-placement puzzle. The object in a 6 by 6 mini-Sudoku puzzle is to fill all empty squares so that the counting numbers 1 to 6 appear exactly once in each row, each column, and each of the 2 by 3 regions, which are delineated by the thick line segments. Solve the following 6 by 6 mini-Sudoku puzzle.The Four 4s Problem The object at this exercise is to create mathematical expressions that use exactly four 4s and that simplify to a counting number from 1 to 20, inclusive. You are allowed to use the following mathematical symbols: +,,,,,(,and). For example, 44+44=2,4(44)+4=5 44+44=18A Cryptarithm The following puzzle is a famous cryptarithm. Each letter in the cryptarithm represents one of the digits 0 through 9. The leading digits, represented by S and M, are not zero. Determine which digit is represented by each of each of the letters so that the addition is correct. Note: A letter that is used more than once, such as M, represents the same digit in each position in which it appears.Determine whether the argument is an example of inductive reasoning or deductive reasoning. All books written by J. K. Rowling make the best-seller list. The book Harry Potter and the Deathly Hallows is a J. K. Rowling book. Therefore, Harry Potter and the Deathly Hallows made the bestseller list.Determine whether the argument is an example of inductive reasoning or deductive reasoning. Samantha got an A on each of her first four math tests, so she will get an A on the next math test.Determine whether the argument is an example of inductive reasoning or deductive reasoning. We had rain each day for the last five days, so it will rain today.Determine whether the argument is an example of inductive reasoning or deductive reasoning. All amoeba multiply by dividing. I have named the amoeba shown in my microscope Amelia. Therefore, Amelia multiplies by dividing.Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers x,x4x.Find a counterexample to show that the following conjecture is false. Conjecture: For all counting numbers n,n3+5n+66 is an even counting number.Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers x,(x+4)2=x2+16.Find a counterexample to show that the following conjecture is false. Conjecture: For numbers a and b, (a+b)3=a3+b3.Use a difference table to predict the next term of each sequence. a. 2,2,12,28,50,78,? b. 4,1,14,47,104,191,314,?Use a difference table to predict the next term of each sequence. a. 5,6,3,4,15,30,49,? b. 2,0,18,64,150,288,490,?A sequence has an nth-term formula of an=4n2n2. Use the nth-term formula to determine the first five terms of the sequence and the 20th term of the sequence.The first six terms of the Fibonacci sequence are: 1,1,2,3,5,and8. Determine the 11th and 12th terms of the Fibonacci sequence.Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figure.Determine the nth-term formula for the number of square tiles in the nth figure.Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. Enclose a Region A rancher decides to enclose a rectangular region by using an existing fence along one side of the region and 2240 feet of new fence on the other three sides. The rancher wants the length of the rectangular region to be five times as long as its width. What will be the dimensions of the rectangular region?Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. True-False Test In how many ways can you answer a 15-question test if you answer each question with either a true, a false or an always false?Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. Number of Skyboxes The skyboxes at a large sports arena are equally spaced around a circle. The 11th skybox is directly opposite the 35th skybox. How many skyboxes are in the sports arena?Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. A Famous Puzzle A rancher needs to get a dog, a rabbit and a basket of carrots across a river. The rancher has a small boat that will only stay afloat carrying the rancher and one of the critters or the rancher and the carrots. The rancher cannot leave the dog alone with the rabbit because the dog will eat the rabbit. The rancher cannot leave the rabbit alone with the carrots because the rabbit will eat the carrot. How can the rancher get across the river with the critters and the carrots?Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. Earnings from Investments An investor bought 20 shares of stock for a total cost of 1200 and then sold all the shares for 1400. A few months later, the investor bought 25 shares of the same stock for a total cost of 1800 and then sold all the shares for 1900. How much money did the investor earn on these investments?Polyas Problem-Solving Strategy Solve each problem using Polya's four-step problem- solving strategy. Label your work so that each of Polyas four steps is identified. Number of Handshakes If 15 people greet each other at a meeting by shaking hands with one another. How many handshakes will take place?Strategies List five strategies that are included in Polya's second step (device a plan).Strategies List three strategies that are included in Polya's fourth step (review the solution).Match Students with Their Major Michael, Clarissa, Reggie, and Ellen are attending Florida State University (FSU). One student is a computer science major, one is a chemistry major, one is a business major, and one is a biology major. From the following clues, determine which major each student is pursuing. a. Michael and the computer science major are next-door neighbors. b. Clarissa and the chemistry major have attended FSU for 2 years. Reggie has attended FSU for 3 years, and the biology major has attended FSU for 4 years. c. Ellen has attended FSU for fewer years than Michael. d. The business major has attended FSU for 2 years.Little League Baseball Each of the Little League teams in a small rural community is sponsored by a different local business. The names of the teams are the Dodgers, the Pirates, the Tigers, and the Giants. The businesses that sponsor the teams are the bank, the supermarket, the service station, and the drugstore. From the following clues, determine which business sponsors each team. a. The Tigers and the team sponsored by the service station have winning records this season. b. The Pirates and the team sponsored by the bank are coached by parents of the players, whereas the Giants and the team sponsored by the drugstore are coached by the director of the community center. c. Jake is the pitcher for the team sponsored by the supermarket and coached by his father. d. The game between the Tigers and the team sponsored by the drugstore was rained out yesterday.27REFind a Route The following map shows the 10 bridges and 3 islands between the suburbs of North Bay and South Bay. a. During your morning workout, you decide to jog over each bridge exactly once. Draw a route that you can take. Assume that you start from North Bay and that your workout concludes after you jog over the 10th bridge. b. Assume you start your jog from South Bay. Can you find a route that crosses each bridge exactly once?Areas of Rectangles Two perpendicular line segment partition the interior of a rectangle into four smaller rectangles. The areas of these smaller rectangles are x,2,5,and10 square inches. Find all possible values of x.Use a Pattern in Make Predictions Consider the following figures. Figure a1 consists of two line segments, and figure a2 consists of four lime segments. If the pattern of adding a smaller line segment to each end of the shortest line segments continues, how many lime segments will be in a. figure a10? b. figure a30?A Cryptarithm In the following addition problem, each letter represents one of the digits 0,1,2,3,4,5,6,7,8,or,9. The leading digits represented by A and B are nonzero digits. What digit is represented by each letter?Make Change In how many different ways can change be made for a dollar using only quarters and/or nickels'?Counting Problem In how many different orders can a basketball team win exactly three out of their laslt five games?34RE35REVerify a Conjecture Use deductive reasoning to show that the following procedure always produces a number that is twice the original number. Procedure: Pick a number. Multiply the number by 4, add 12 to the product, divide the sum by 2, and subtract 6.Explain why 2004 nickels are worth more than 100.Gasoline Prices The following bar graph shows the average U.S. unleaded regular gasoline prices for the years from 2007 to 2013. a. What was the maximum average price per gallon during the years from 2007 to 2013? b. During which two consecutive years did the largest price increase occur?Super Bowl Ad Price The following graph shows the price for a 30-second Super Bowl ad from 2008 to 2015. a. What was the price of a 30-second Super Bowl ad in 2013? Round to the nearest tenth of a million dollars. b. In 2015, about 11.8 million people watched the Super Bowl on television. What was the price per viewer that was paid by an advisor that purchased a 4.5 million ad? Round to the nearest cent per viewer.Search Engine Rankings The following circle graph shows the percent of the 18.6 billion U.S. searches that were conducted in August 2015 by the top five search engines. a. How many searches were conducted by Google in August 2015? Round to the nearest tenth of a billion. b. How many times more searches were conducted by Yahoo than by ASK in August 2015? Round to the nearest tenth.Palindromic Numbers Recall that palindromic numbers read the same from left to right as they read from right to left. For instance, 37,573 is a palindromic number. Find the smallest palindromic number larger than 1000 that is a multiple of 5.Narcissistic Number A narcissistic number is a two digit natural number that is equal to the sum of the squares of its digits. Find all narcissistic numbers.Number of Intersections Two different lines can intersect in at most one point. Three different lines can intersect in at most three points, and four different lines can intersect in at most six points. a. Determine the maximum number of intersections for five different lines. b. Does it appear, by inductive reasoning that the maximum number of intersection points In for n different lines is given by In=n(n1)2?44REA Numerical Pattern A student has noticed the following pattern. 91=9 has 1 digit. 92=81 has 2 digits. 93=729 has 3 digits. 910=3,486,784,401 has 10 digits. a. Find the smallest natural number n such that the number of digits in the decimal expansion of 9n is not equal to n. b. A professor indicates that you can receive five extra-credit points if you write all of the digits in the decimal expansion of 9(99). Is this a worthwhile project? Explain.Inductive vs. Deductive Reasoning Determine whether the argument is an example of inductive reasoning or deductive reasoning. Two computer programs, a bubble sort and a shell sort, are used to sort data. In each of 50 experiments, the shell sort program took less time to sort the data than did the bubble sort program. Thus the shell sort program is the faster of the two sorting programs.Inductive vs. Deductive Reasoning Determine whether the argument is an example of inductive reasoning or deductive reasoning. If a figure is a rectangle, then it is a parallelogram, Figure A is a rectangle. Therefore, Figure A is a parallelogram.Use a difference table to predict the next term in the sequence 1,0,9,32,75,144,245,....List the first 10 terms of the Fibonacci sequence.In each of the following, determine the nth-term formula for the number of square tiles in the nth figure.A sequence has an nth-term formula of an=(1)n(n(n1)2) Use the nth-term to determine the first 5 terms and the 105th term in the sequence.Terms of a Sequence In a sequence: a1=3,a2=7 and an=2an1+an2 for n3 Find a3, a4 and a5.Number of Diagonal A diagonal of a polygon is a line segment that connects nonadjacent vertices (corners) of the polygon. In the following polygons, the diagonals are shown by the blue line segments. Use a difference table to predict the number of diagonals in a. a heptagon (a 7-sided polygon) b. an octagon (an 8-sided polygon)State the four steps of Polyas four-step problem-solving strategy.10TCounting Problem In how many different ways can a basketball team win exactly four out of their last six games?Units Digit What is the units digit (ones digit) of 34513?Vacation Money Shelly has saved same money for a vacation Shelly spends half of her vacation many on an airline ticket; then spends 50 for sunglasses, 22 for a taxi, and one-third of her remaining money for a room with a view. After her sister repays her a loan of 150, Shelly finds that she has 326. How much vacation money did Shelly have at the start of her vacation?Number of Different Routes How many different direct routes are there from point A to point B in the following figure?Number of League Games In a league of nine football teams, each team plays every other team in the league exactly once. How many league games will take place?Ages of Children The four children in the Rivera family are Reynaldo, Ramiro, Shakira and Sasha. The ages of the two teenagers are 13 and 15. The ages of the younger children are 5 and 7. From the following clues, determine the age of each of the children. a. Reynaldo is older than Ramiro. b. Sasha is younger than Shakira. c. Sasha is 2 years older than Ramiro. d. Shakira is older than Reynaldo.Counterexample Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers x, (x4)(x+3)x4=x+3.Counterexample Find a counterexample to show that the following conjecture is false. Conjecture: For all numbers x, xx2.Find a Sum Find the following sum without using a calculator. 1+2+3+4+...+497+498+499+500Motor Vehicle Thefts The following graph shows the number of U.S. motor vehicle thefts for each year from 2009 to 2014. a. Which one of the given years had the greatest number of U.S. motor vehicle thefts? b. How many more U.S. motor vehicle thefts occurred in 2011 than in 2013? c. During which two consecutive years did the largest decline in motor vehicle thefts occur?Mark, Erica, Larry, and Jennifer have each defined a fuzzy set to describe what they feel is a good grade. Each person paired the letter grades A, B, C, D. and F with a membership value. The results are as follows. Mark: M = {(A, 1), (B, 0.75), (C, 0.5), (D, 0.5), (F, 0)} Erica: E = {(A, 1), (B, 0), (C, 0), (D, 0), (F, 0))} Larry: L={(A, l), (B, l), (C, l), (D, I), (F,0)} Jennifer: J — ((A, 1), (B, 0.8), (C, 0.6), (D, 0.1), (F, 0)) a. Which of the four people considers an A grade to be the only good grade! b. Which of the four people is most likely to be satisfied with a grade of D or better? c. Write a fuzzy set that you would use to describe the set of good grades. Consider only the letter grades A, B, C, D, and F.In some fuzzy sets, membership values are given by a membership graph or by a formula. For instance, the following figure is a graph of the membership values of the fuzzy set OLD. Use the membership graph of OLD to determine the membership value of each of the following. a. x = 15 b. x = 50 c. x = 65 d. Use the graph of OLD to determine the age x with a membership value of 0.25. An ordered pair (x, y) of a fuzzy set is a crossover point if its membership value is 0.5.e. Find the crossover point for OLD.The following membership graph provides a definition of real number x that are 4about 4. Use the graph of ABOUTFOUR to determine the membership value of: a. x = 2 b. x = 3.5 c. x=7 d. Use the graph of ABOUTFOUR to determine its crossover points.The membership graphs in the following figure provide definitions of (he fuzzy sets COLD and WARM. The point (35, 0.5) on the membership graph of COLD indicates that the members hip value for x = 35 is 0.5. Thus, by this definition, 35°F is 50% cold. Use the above graphs to estimate a. the WARM membership value for x 40. b. the WARM membership value for x = 50. c. the crossover points of WARM.The membership graph in Excursion Exercise 2 shows one persons idea of what ages are old. Use a grid similar to the following to draw a membership graph that you feel defines the concept of being young in terms of a persons age in years. Show your membership graph to a few of your friends. Do they concur with your definition of young?1ES2ES3ES4ES5ES6ES7ES8ES9ES10ES11ES12ESIn Exercises I to 14, use the roster method to write each of the given sets. For some exercises you may need to consult a reference, such as the Internet or an encyclopedia. The set of letters in the English alphabet, except the letter y, that are vowels14ES15ESIn Exercises 15 to 24, write a word description of each set. There may be more than one correct description. {Libra, Leo}In Exercises 15 to 24, write a word description of each set. There may be more than one correct description. {Mercury, Venus}In Exercises 15 to 24, write a word description of each set. There may be more than one correct description. {penny, nickel, dime}19ES20ES21ES22ES23ES24ESIn Exercises 25 to 36, determine whether each statement is true or false. If the statement is false, give a reason. b {a, b, c}26ES27ES28ES29ESIn Exercises 25 to 36, determine whether each statement is true or false. If the statement is false, give a reason. The set of large numbers is a well-defined set.31ES32ES33ES34ES35ES36ES37ES38ES39ES40ESIn Exercises 37 to 48, use set-builder notation to write each of the following sets. {January, March, May, July, August, October, December}42ES43ES44ES45ES46ES47ES48ES49ES50ES51ES52ES53ES54ES55ES56ES57ES58ES59ES60ES61ES62ES63ESIn Exercises 63 to 70, state whether each of the given pairs of sets are equal, equivalent, both, or neither. The set of single-digit natural numbers; the set of pins used in a regulation bowling gameIn Exercises 63 to 70, state whether each of the given pairs of sets are equal, equivalent, both, or neither. The set of positive whole numbers; the set of natural numbers66ES67ES68ES69ES70ES71ES72ES73ES74ES75ES76ES77ES78ES79ES80ES