Mathematics in the Modern World PDF

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1 PELAYO, KRISTINA S. BSA -2B

MATHEMATICS IN THE MODERN WORLD MODULES CHAPTER 1 – CHAPTER 5

CHAPTER 1 Exercise 1.1 Write an essay about how you use Mathematics in our world using the following guided questions: (at least 150 words) o What is mathematics for you? o Where do you apply the principles of mathematics? o Do you need mathematics every day? Why? o What have you learned from school on mathematics so far? o Do you appreciate mathematics? Why or why not? For a student like me who’s not so into Mathematics, it is something that’s around us, everywhere we go, even though we’re not aware of it. One definition of Mathematics that retained in my mind (which I learned when I was in junior high school) is that it’s the building block for everything in our daily lives. In my Accounting course, I always use the four fundamental operations and my analytical skills in solving problems. I also love baking pastries and I often do conversion of ingredient measures. I am also addicted to online shopping so I often do budgeting of my allowance, forecasting when to order so I could expect the arrival of my order and such. Surely I need Mathematics in my daily life. It answers most of my questions daily like how many hours do I have left to advance read for my next class, what time should I finish this so I can do the other activities of other subjects, should I order a book now or I’ll wait for my money next month so I could order a bulk to save in shipping fee, and many more. As far as I can remember in my junior high school, I’ve learned about statistics and probabilities, basic algebra, the Phytagorean Theorem, trigonometric functions, and I’m sure there’s more, I just forgot about it since it’s been a while. I may not be a Mathematic genius but I appreciate it. It’s amusing how Mathematics can provide you with certain answers most of the time, how it affects the decision-makings of people every day, how it can give you a projection of the future. Mathematics is just incredibly interesting; it’s the opposite of how most people perceived it as a boring and complicated subject. And to just add, I feel proud knowing that at a point in my life (high school days) though, I was able to solve such complicated Math problems. Exercise 1.2

Identify at least 10 patterns and regularities in your surroundings by taking photos, describe each by applying principles of mathematics.

Fibonacci effect is present in the unfurling rose petals, cactus, and succulents.

The shells and human ear follows the Fibonacci Spiral.

Spiders build its web according to its specie.

2

Fibonacci sequence is present in petals of Yellow Bell, Santan, and tangkay of Malunggay and Tamarind leaves. Exercise 1.3 1. With your partner, measure each of the following: Partner: Raymond Lima Unit of length: Centimeter Kristina Pelayo Actual What should be o Height 146.50 146.50 = 1.7235 = 1.618 Height of navel 85 90.54 o Foot 23 23 = 1.3529 = 1.618 Hand length 17 14.215 o Length of forearm 22 22 = 1.2941 = 1.618 Length of hand 17 13.597 o Width of center tooth 0.9 0.9 = 1.8000 = 1.618 Width of second tooth 0.5 0.5562 o Shoulder length 89 89 = 1.2535 = 1.618 Waistline 71 55

Actual

Raymond Lima What should be

2. A wood that is 12 feet in length is needed to be cut into two parts such that the ratio of the parts constitutes the golden ratio. What must be the lengths of the wood?

Long Part Short Part

7.4166 4.5834

12__ _ 7.4166

= 1.618

3

Exercise 2.1

CHAPTER 2 Write an essay about the Language of Mathematics using the following guided questions: (at least 150 words) o Is language of Mathematics important to you? Why or why not? o When do you use the language of Mathematics? o Can you live without it? Why or why not?

The language of Mathematics is important to me especially now that my course is Accountancy and that my future career will have me dealing with numbers. Even if I am not taking this course, the Mathematics language will still remain crucial as I am using it every day. I deal with simple Maths every day so learning the basics is necessary. As I’ve said from Chapter 1, I use Mathematics as in the whole time of the day! And so as it’s language since it’s the way I can comprehend what I am analyzing and the things that I need to know. I cannot escape Math so living without it is impossible. At the time a baby could speak, it’s what’s mostly taught, how to count. Mathematics and it’s language, whether I like it or not, is part of my life. And I welcome Math in my life, open arms. Exercise 2.2

Tell whether if each of the following sentences is an open sentence or a closed sentence an. Write OS if a sentence is open and CS if it is closed. If CS, determine if it is true or false. If OS , identify the expression that will make the sentence always true.

CS (FALSE) CS (TRUE) CS (FALSE) OS (n) CS (TRUE)

1. 3. 5. 7. 9.

Nine is an even number. Zero is an even number. ½>⅔ 2n < 5 ℼ is a variable.

OS (x) OS (x) OS (n) CS (TRUE) CS (TRUE)

2. 4. 6. 8. 10.

4x – 2 = 5 2 + 5 = 2x n is a composite number. -0.5 is an integer. 0 is not an integer.

Exercise 2.3 A. Translate each of the following English sentences into Mathematical sentences. 1. The square of the difference of x and y is not more than 10.  (x-y)2 < 10 2. The square of a number is positive.  √x 3. Four is an even number.  4 ∈ {2𝑛, 𝑛 ∈ ℕ} 4. One-fourth is a rational number.  1/4 ∈ Q 5. Six is the principal square root of 36.  √36 = 6 B. Translate each of the following Mathematical sentences into English sentences. 1. ∀𝑥 ∈ ℝ, ∃𝑦 ∈ ℝ, 𝑥 + 𝑦 = 10  For any real number, there exists a real number y such that x + y equals ten. 2. ∀𝑥 ∈ ℤ+, ∃𝑦 ∈ ℝ, 𝑦2 = x  For any positive integer, there exist a real number y such that 𝑦2 equals x. 3. 𝑥 + 12 = 8  The sum of x and twelve is eight.

4 4. 2(𝑥 − 3) = 12  Two times the difference of x and three is twelve. 5. 2𝑥 − 6 = 45  Two times a number x minus six is forty-five. CHAPTER 3 Exercise 3.1 A. Determine if each statement is a proposition or not. PROPOSITION 1. Every triangle is a polygon. PROPOSITION 2. All right angles are congruent. PROPOSITION 3. x is greater than or equal to -2. NOT A PROPOSITION 4. If x + 2 = 4, is x = 2? NOT A PROPOSITION 5. The sum of the interior angles in a triangle. PROPOSITION 6. Some rectangles are not parallelograms. PROPOSITION 7. Each equilateral triangle is an isosceles triangle. PROPOSITION 8. For all values of a and b, (a + b)(a – b) = a 2 – b2 PROPOSITION 9. If a is a real number, a 2> 2. NOT A PROPOSITION 10. Bisect an angle. B. Determine whether each statement is A, E, I, or O proposition. I 1. Some variables are fractions. O 2. Each scalene triangle has no equal sides. I 3. Some rectangles are parallelograms. A 4. All right angles are congruent. E 5. Every triangle is not a polygon. I 6. Few rational numbers are integers. A 7. Every odd number is prime. O 8. Some irrational numbers are not terminating decimals. A 9. All mathematicians are males. O 10. Some polynomials are not congruent sides. Exercise 3.2 I. Write each statement in symbolic form using connectives ¬, ˅, ˄, →, or ↔. 1. If today is Friday (p), then tomorrow is Saturday (q). p→q p˄q 2. I went to the registrar’s office (p) and I ate lunch at the canteen (q). 3. A triangle is an equilateral triangle (p) if and only if it is an equiangular triangle (q). p↔q 4. If it is bird (p), then it has feathers (q) p→q 5. If either x is a fraction or y is a decimal (p), then it is not a rational number (¬q). p→q 6. The moon is flat (p) if and only the sun rises at the south (q) and the dog is flying (r). p ↔ (q ˄ r) (p ˅ ¬q) → r 7. If either a frog is an amphibian (p) or a jelly fish is not a fish (¬q), then 1 + 2 = 3 (r). ¬p ↔ (¬q ˄ r) 8. Online games are not bad to students (¬p) if and only if it will not destroy their studies (¬q) and they sleep at least 8 hours a day (r). 9. Cigarette smoking is dangerous to your health (p) and it gives bad breath (q), or it can kill you (r). (p ˄ q) ˅ r (p ˄ q) → r ˅ s 10. If x + 5 = 7 (p) and 2x – 7 = 6 (q), then x + 8 = 2 (r) or 6x – 2 = 4 (s).

5 II.

Write each symbolic statement as an English sentence. Use p, q, r, s, and t as defined below. p: Sarah Geronimo is a singer. q: Sarah Geronimo is not a songwriter. r: Sarah Geronimo is an actress. s: Sarah Geronimo plays guitar. t: Sarah Geronimo is a dancer.

1. (p ˅ r) ˄ q  Sarah Geronimo is a singer or an actress, and she is not a songwriter. 2. p → (q ˄¬r)  If Sarah Geronimo is a singer, then she is not a songwriter and not an actress. 3. (r ˄ p) ↔ q  Sarah Geronimo is an actress and a singer if and only if she is not a songwriter. 4. ¬s → (p ˄¬q )  If Sarah Geronimo does not play guitar then she is a singer and a songwriter. 5. 𝑡 ↔ (¬r ˄¬p)  Sarah Geronimo is a dancer if and only if she is not an actress and she is not a singer. Exercise 3.3 I. Write each sentence in symbolic form. Use p, q, r, and s as defined below. Tell whether if each statement is true or false by applying the truth table. p: Stephen Curry is a football player. (False) q: Stephen Curry is a basketball player. (True) r: Stephen Curry is a rock star. (False) s: Stephen Curry plays for the Warriors. (True) (p ˅ q) ˄ ¬r r ˄ (¬q ˅ p) (q ˄ r) → ¬p q ↔ (¬p ˄ ¬r) s → (q ˄ ¬p) II. p T T T T F T T T F F F T F F F F

q T T T F T T F F T T F F T F F F

1. 2. 3. 4. 5.

Stephen Curry is a football player or a basketball player, and he is not a rock star. Stephen Curry is a rock star, and he is not a basketball player or a football player. If Stephen Curry is a basketball player and a rock star, then he is not a football player. Stephen Curry is a basketball player, if and only if he is not a football player and he is not a rock star. If Stephen Curry plays for the Warriors, then he is a basketball player and he is not a football player.

Construct a truth table for [(¬𝑝 ∨ 𝑞 ∨ 𝑟)] ∧ [𝑠 ∧ (¬𝑞 ∨ ¬𝑟)] 𝑞 ∨ ¬𝑟𝑟 ) 𝑝 ∨ 𝑞 ∨ 𝑟 ) (¬𝑞 𝑞 ∨ ¬𝑟 ) 𝑠 ∧ (¬𝑞 r s ¬p ¬q ¬r (¬𝑝 T T F F F T F F T F F F F T F F F T F F T F T T T T F T F F T T T T T F F T F F F F F F T F T F F T F T T F T T T F F T F F T F F T T F T T T T T F T F F T F F T T T T F T T T F F F T T F T F F F T F T T T F T F T T F T T F F T T T T F T T F F T T T F T F

[(¬𝑝 𝑝 ∨ 𝑞 ∨ 𝑟 )] ∧ [𝑠 ∧ (¬𝑞 𝑞 ∨ ¬𝑟𝑟 )] F F F F F F F F T F T F F F F F

6 CHAPTER 4 Exercise 4.1 A. Use inductive reasoning to predict the next number in each of the following lists. 1. 3, 6, 9, 12, 15, 18 2. 1, 3, 6, 10, 15, 21 3. 2, 4, 8, 16, 32, 64, 128 4. 1, 8, 27, 64, 125, 216 5. 2, 5, 10, 17, 26, 37 B. Use Inductive Reasoning to make a conjecture. Complete the procedure for several different numbers. Consider the following procedure: 1. 2. 3. 4. 5.

Pick a number. Multiply the number by 9. Add 15 to the product. Divide the sum by 3. And subtract 5.

1 1×9=9 9 + 15 = 24 24 ÷ 3 = 8 8–5=3

2 2 × 9 = 18 18 + 15 = 33 33 ÷ 3 = 11 11 – 5 = 6

3 3 × 9 = 27 27 + 15 = 42 42 ÷ 3 = 14 14 – 5 = 9

4 4 × 9 = 36 36 + 15 = 51 51 ÷ 3 = 17 17 – 5 = 12

5 5 × 9 = 45 45 + 15 = 60 60 ÷ 3 = 20 20 – 5 = 15

C. Verify that each of the following statement is a false statement by finding a counterexample for each. For all numbers X: 1. = 1 

Let x = 0. Then is indeterminate. “For all numbers x, = 1” is a false statement. =x+1

2.  3. √𝑥 

Let x = 2. Then

= 2 + 1. “For all numbers

= x + 1” is a false statement.

=x+ 4 Let x = 3. Then √

= 3 + 4. “For all numbers √𝑥

= x + 4” is a false statement.

D. Use Inductive Reasoning to Solve an Application Scientists often use inductive reasoning. For instance, Galileo Galilei (1 564-1642) used inductive reasoning to discover that the time required for a pendulum to complete the swing, called the period of the pendulum, depends on the length of the pendulum. Galileo did not have a clock, so he measured the periods of pendulum in “heartbeats.” The following table shows some results obtained for pendulums of various lengths. For the sake of convenience, a length of 10inches has been designated as 1 unit.

Use the data in the above table and inductive reasoning to answer each of the following questions. a. If a pendulum has a length of 49 units, what is its period?  A pendulum that has a length of 49 units will have a 7 period in heartbeats. b. If the length of a pendulum is quadrupled, what happens to its period?  If the length of a pendulum is quadrupled, its period doubles because the frequency goes down by a factor of 2.

7 E. Use Deductive Reasoning to Establish a Conjecture 1. Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Solution: Let k represent the original number.  Multiply k by 8 8k  Add 6 to the product 6 + 8k  Divide the sum by 2 (6 + 8k) ÷ 2 = 3 + 4k  And subtract 3 3 + 4k – 3 = 4k  This means that the procedure produces a number that is four times larger than the original number. 2. Consider the following procedure: Pick a number. Add 3 to the number and multiply the sum by 2. Subtract 6 from the product then divide the result by 2. Solution: Let r represent the original number.  Add 3 to the number r+3  Multiply the sum by 2 (r + 3) × 2 = 2r + 6  Subtract 6 from the product 2r + 6 – 6 = 2 r 

Divide the result by 2



This means that the procedure produces the original number.

=r

F. Each of four neighbors, Sean, Maria, Sarah, and Brian, has a different occupation (editor, banker, chef, and dentist). From the following clues, determine the occupation of each neighbor. 1. 2. 3. 4.

   

Maria gets home from work after the banker but before the dentist. Sarah, who is the last to get home from work, is not the editor. The dentist and Sarah leave for work at the same time. The banker lives next door to Brian. EDITOR BANKER CHEF DENTIST x3 x3 x4 SEAN  x1 x3 x1 MARIA   x2 x2 x3 SARAH  x3 x4 x3 BRIAN From Clue #1, Maria is neither the banker nor the dentist. From Clue #2, Sarah is not the editor. If Sarah is the last to get home, then she’s not the banker from #1. From Clue #3, Sarah is not the dentist, therefore she is the chef. Maria cannot be a chef anymore so she is the editor. Neither Sean nor Brian can be the editor or chef anymore. From Clue #4, Brian is not the banker, so he is the dentist. Lastly, Sean is the banker.

Exercise 4.2 A. Use a difference table to predict the next term in the sequence. 1. 1, 14, 51, 124, 245, 426, 679 2. -2, 2, 12, 28, 50, 78, 112 3. -4, -1, 14, 47, 104, 191, 314, 479 4. 5, 6, 3, -4, -15, -30, -49, -72 5. 2, 0, -18, -64, -150, -288, -490, -768

8 B. Use the given nth term formula to compute the first six terms of the sequence. 1. an = 2-n 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625 2. an = (-1) n+1n2 1, -4, 9, -16, 25, -36 0, 1.5, 2.67, 3.75, 4.8, 5.83

3. an = 4. an =

0.5, 0.67, 0.75, 0.8, 0.83, 0.86

5. an = (-1) (n – n + 7) -7, -9, -13, -19, -27, -37 C. Expand the following algebraic expressions using Pascal’s Triangle. 1. (x + y) 5 32x10 – 80x8y3 + 80x6y6 – 40x4y9 + 10x2y12 – y15 4 2. (x – 2y) x4 – 8x3y + 24x2y2 – 32xy3 + 16y4 3. (x + y)8 x8 + 8x7y + 28x6y2 + 56x5y3 + 70x4y4 + 56x3y5 + 28x2y6 + 8xy7 + y8 4 4. (3x + 2y) 81x4 + 216x3y + 216x2y2 + 96xy3 + 16y4 2 3 5 5. (2x – y ) 32x10 – 80x8y3 + 80x6y6 – 40x4y9 + 10x2y12 – y15 2

D. Determine the minimum number of moves required to transfer all of the disks to another peg for each of the following situations. 1. You start with four disks. 15 moves 2. You start with five disks. 31 moves 3. You start with six disks. 63 moves 4. You start with seven disks. 127 moves 5. You start with eight disks. 255 moves Exercise 4.3 A. Apply the Polya’s Problem Solving Strategy by identifying your own problem and life. Problem: I am having a hard time answering my Math module. Understand the problem.  I keep rereading the theories and the examples so I can figure out what to do. Devise a plan.  I’ll get scratch papers and I’ll try to do the same what the examples show. Carry out the plan.  Solving the problems in accordance with the examples. Review the solution.  Resolving to make sure that I did the right procedures and that I arrived at the right answer. B. Apply Polya’s Problem Solving Strategy (Guess and Check) 1. A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games?  There are six different orders they could win. WWLL WLWL WLLW

LLWW LWLW LWWL

9 2. Determine the digit 100 places to the right of the decimal point in the decimal representation 7/27.  7/27 is 0.259. Then, 99 is a multiple of three so it will end up at number 9. Therefore, the 100 th digit is 2. 3. The product of the ages, in years, of three teenagers is 4590. None of the teens are the same age. What are the ages of the teenagers?  The product of the three ages is a multiple of 10, thus, also a multiple of 5. We could use 15 from the list of teen ages and divide it to 4,590. 4,590 ÷ 15 = 306  Looking at the teen ages which end digit when multiplied is 6, it’s 17 and 18. 15 × 17 × 18 = 4,590 4. A hat and a jacket together cost Php 100.00. The jacket costs Php 90.00 more than the hat. What are the cost of the hat and the cost of the jacket?  First, take away the Php 90.00 cost of the jacket and we’re left with Php 10.00. Divide it by 2, equals 5. Add the Php 90.00 to the other 5 so it’s Php 95.00. Jacket = Php 90.00 + 5.00 = Php 95.00 Hat = Php 5.00 CHAPTER 5 Exercise 5.1.1 1. Mrs. Rodriguez wants to purchase a washing machine listed at ₱25,000 cash and ₱25,950 if paid at an instalment basis of 4 months. What is the rate of interest?  11.4% 2. An interest of ₱850 was earned in 5 months on an investment at 10%. How much was invested?  ₱20,400 3. What principal will accumulate to ₱215,000 in 3 years at 12% simple interest?  ₱597,222.22 4. A bank issued a 6-year loan of ₱500,000 with a simple interest of 7% to an employee. Determine the interest which the employee must pay.  ₱210,000 5. A cash of ₱250,000 is deposited to an account paying at 5% simple interest. How much is the account after five years?  ₱312,500 6. Find the interest on a loan of ₱65,000 at 12% interest which will be paid after 6 months.  ₱3,900 7. A ₱10,000 savings account earned ₱1,400 interest in 3 years. What was the rate of interest given?  4.67% 8. Find the number of days from March 15 to September 15 of the same year and calculate the simple interest due on a ₱35,800 loan made with an interest rate of 1.5%.  Exact Method: ₱270.71 Ordinary Method: ₱274.47 9. Calculate the simple interest due on a ₱25,400 loan made on June 30 and repaid on February 25 of the following year with 1.65% given interest rate.  Ordinary Method: ₱279.40 Exact Method: ₱275.57 10. Find the due date on a 60-day loan made on November 11.  January 10

10

Exercise 5.1.2 1. Calculate the maturity value of a simple interest, a 10-month loan of ₱20,000 if the interest rate is 3.75%.  ₱20,625 2. A credit union has issued a 6-month loan of ₱10,500 at a simple interest rate of 2.5%. What amount will be repaid at the end of six months?  ₱10,631.25 3. An employee applied a ₱50,000 loan from the bank. If she agrees to pay the loan in 6 months with a simple interest rate of 1.25% per month. How much should he repay the bank?  ₱50,312.50 4. ₱45,000 is borrowed for 90 days at a 5% interest rate. Calculate the maturity value by the exact method and by the ordinary method.  Exact Method: ₱45,554.79 Ordinary Method: ₱45,562.50 5. Joshua borrowed ₱4,895 from his employer. He promised to repay him in 60 days with an interest of 10%. How much will he pay using the exact interest?  ₱4,975.47 Exercise 5.1.3 1. Find ...