Math 1100 Module 1 PDF

MATH 1100 Department of Mathematics and Physics MODULE 11 Nature of Mathematics: Mathematics in our World Overview Patterns in nature are regularities that can be observed a...

MATH 1100 Department of Mathematics and Physics MODULE 11 Nature of Mathematics: Mathematics in our World Overview Patterns in nature are regularities that can be observed around us. In human history, it is perhaps the ancient Greeks who are the most known in studying patterns in their attempt to explain regularities in nature. Examples of these patterns are tessellations, spots, stripes and symmetries. In this module, you will be introduced to the different patterns and numbers seen in nature. Other patterns and concepts like fractals, Fibonacci sequence, the Euler number e and its application to population growth are discussed. Time allotment: 2 weeks Objectives: Upon completion of this module, you are expected to: 1. identify the patterns in nature and regularities in the world 2. articulate the importance of Mathematics in one’s life. 3. argue about the nature of Mathematics, what it is, how it is expressed, represented and used. 4. express appreciation for Mathematics as a human endeavor. PRE-ASSESSMENT Before engaging yourself in the following discussions, I want you to share your ideas about the following questions: 1. Do you see any patterns in your place? Describe these patterns. 2. How do you differentiate these patterns from one another? 3. Do these patterns exhibit symmetry? If yes, what are these symmetries? 1 This module is based from the book “Mathematics in the Modern World” by the Department of Mathematics and Physics, CS, CLSU. 1  MATH 1100 Department of Mathematics and Physics 1. PATTERNS IN NATURE Patterns in nature are regularities that can be observed around us. In human history, it is perhaps the ancient Greeks are the most known in studying patterns in their attempt to explain regularities in nature. Describing regular forms involves studying their symmetries. In the following sections, you will be introduced to the concept of symmetry and the different kinds of symmetry. 1.1 Symmetry The word symmetry comes from the Greek word symmetria, meaning “the same measure”. A symmetry of an object in the plane is a rigid motion of the plane that leaves the object unchanged. Intuitively, why does the spade in Figure 1a looks symmetric? Answer: It looks the same from many positions. It has repeated parts. (a) (b) Figure 1. (a) Spade and (b) its repeated part colored gray. The line of symmetry is colored red. A repeated part of the spade in Figure 1 is colored gray. Flipping the spade along the red line in Figure 1b leaves the spade unchanged. So we say the flipping motion is a symmetry of the spade. This flipping motion is called reflectional symmetry (sometimes called bilateral or mirror symmetry). 1.1.1 Reflectional Symmetry How to know that an object has reflectional symmetry? Answer: If you fold a picture in half and both halves are exact mirror image of one another, then the figure has a reflectional symmetry. The fold is what we call the line or axis of symmetry. 2  MATH 1100 Department of Mathematics and Physics In Figure 1, the line of symmetry of the spade is colored red. Self-Assessment Question 1 (SAQ1). The human body, animals like crabs, and insects like spiders exhibit reflectional symmetries (Figure 2). Can you tell where their respective line of symmetry. Figure 2. Human body, crab and spider with their respective lines of symmetry. Example 2. The flower with four petals in Figure 3 has reflection symmetries. It has 4 lines of symmetry. (a) (b) Figure 3. (a) A flower having multiple lines of symmetry. (b) The 4 lines of symmetry of the flower with four petals in (a). 3  MATH 1100 Department of Mathematics and Physics SAQ2. Can you tell if the objects in Figure 4 have reflectional symmetries? How many lines of symmetry do they have? (a) (b) Figure 4. (a) Starfish; (b) Recycle symbol. 1.1.2 Rotational Symmetry Rotational or radial symmetry is a rigid motion that makes an object look exactly the same as it did before it was rotated about a fixed point, called the center. The rotation must be less than 3600. The number of times an object can be rotated is called the order; and, the number of degrees through which an object is rotated so that it still looks the same as it did before the rotation, is called the angle of rotation. 360𝑜 Let 𝑛 be the order then the smallest angle of rotation is given by 𝑛. Example 3. The recycle symbol in Figure 4b has a rotational symmetry. Its center of rotation is colored red shown in Figure 5a. How to know the order? Consider an arrow-like part and color it blue. After the first rotation, it will be moved to the next arrow-like part. You need 3 rotations to arrive at the starting position. (See Figure 5). Its order is 3. The smallest angle of rotation is 360°/3 = 120°. We say that it has 120° rotational symmetry. It also has 2(120°) = 240° rotational symmetry and 360° = 0° rotational symmetry. 4  MATH 1100 Department of Mathematics and Physics (a) (b) (c) (d) starting position after 3 rotations Figure 5. Determining the order of rotation in the recycle symbol. (b) Obtained by applying 120° rotational symmetry to (a). (c) Obtained by applying 240° rotational symmetry. (d) Obtained by applying 360° = 0° rotational symmetry SAQ3. Does the flower with four petals in Figure 3 has rotational symmetry? How about the starfish in Figure 4a? What is their respective order of rotation? What is their smallest angle of rotation? What are their rotational symmetries? 1.1.3 Translation Symmetry Translations are transformations that slide objects along without rotating them (Stewart, 1995). We say that a pattern has a translation symmetry if an object in the pattern has been moved the same distance and the same direction. A translation also preserves orientation. Example 4. Foot prints of kangaroo shows translation symmetries. (See Figure 6a). If you stretch a sea snake in a line, the pattern in its skin also exhibits translation symmetries. (See Figure 6b). (a) (b) Figure 6. (a) Footprints of a kangaroo; (b) A sea snake2. SAQ4. What other patterns in nature exhibits translation symmetry? Can you give one? 2 Jens Petersen, Image of a banded sea krait, Laticauda colubrina. Taken at Lembeh Straits, North Sulawesi, Indonesia. Taken from https://en.wikipedia.org/wiki/Sea_krait#/media/File:Laticauda_colubrina_Lembeh2.jpg. 5  MATH 1100 Department of Mathematics and Physics 1.1.4 Spirals A spiral is formed because of a property of growth known as self-similarity or scaling, which means that the same shape is maintained (not of the same size) as the object (creature) grows. Example 5. Examples of objects with spirals are shown in Figure 7. Figure 7a shows a satellite image when air spirals towards the eye of the typhoon in a low pressure system. Both the nautilus shell in Figure 7b and the Dall sheep horn in 7c exhibit logarithmic spiral (can be describe using logarithmic function). (a) (b) (c) Figure 7. Spirals in nature: (a) Satellite image of a typhoon3; (b) Cross-section of a nautilus shell4; (c)The horns of a Dall Sheep5 SAQ5. What other objects in nature exhibit spiral pattern? Can you give one? 1.2 Tessellations Tessellation (or tiling) is a pattern made up of one or more geometric shapes that are joined together without overlaps or gaps to cover a plane. Example 6. Examples of tessellations in nature are shown in Figure 8. Tessellations can be observed from the honeycombs of bees, the snake skin laid down on a plane, the shapes formed by the veins on a leaf, the cracked mud where the cracks are considered as lines and not as gaps, and the patterns formed by tides on the sand. 3 Taken from https://www.pinterest.ph/pin/132363676520672456/ 4 Taken from www.ratemyscreensaver.com 5 Taken from http://www.patternsinnature.org/Book/Spirals.html 6  MATH 1100 Department of Mathematics and Physics (a) (b) (c) (d) (e) Figure 8. Tessellations in nature: (a) Honeycomb ; (b) snake’s skin7; (c) leaf8; (d) 6 cracked mud9; (c) tidal10; SAQ6. What symmetries can be observed on tessellations? Are there always symmetries on them? 1.3 Fractals A fractal is a never ending replication of a pattern at different scales (same shape but different size). This property is called self-similarity. Example 7. Among the known fractal is the von Koch curve, named after its creator Neils Fabian Helge von Koch. The method to create this curve is to start with a single line segment. Divide the line segment into three equal parts, remove the middle part and replace it with the two sides of an equilateral triangle of length equal to the length of the segment that has been removed. Then repeat the process to each of the resulting line segments. Figure 9 shows the first four iterations of von Koch curve. 6 Taken from http://www.spacemakeplace.com/wp-content/uploads/2015/07/Honeycomb_pattern.jpg 7 Taken from http://www.spacemakeplace.com/wp-content/uploads/2015/07/Snake_pattern.jpg 8 Taken from http://www.spacemakeplace.com/wp-content/uploads/2015/07/Leaf_pattern.jpg 9 Taken from http://www.spacemakeplace.com/wp-content/uploads/2015/07/CrackedMud_pattern.jpg 10 Taken from http://www.spacemakeplace.com/wp-content/uploads/2015/07/Tidal_pattern.jpg 7  MATH 1100 Department of Mathematics and Physics Figure 9. First four iterations of von Koch curve.11 SAQ7. Figure 10 shows the first three iterations Minkowski curve. How is the minkowski curve constructed? Figure 10. First three iterations of Minkowski curve.12 Example 8. Fractals in nature can be observed from the forming of rivers (see Figure 11a) and from the forming of ice crystals (see Figure 11b). Even on a broccoli, fractals can also be observed (see Figure 11c). 11 Figure taken from www.researchgate.net/figure/Koch-curve-at-iterations-from-0-to-4_fig1_316742654 12 Figure taken from www.researchgate.net/figure/Koch-curve-at-iterations-from-0-to-4_fig1_316742654 8  MATH 1100 Department of Mathematics and Physics (a) (b) (c) Figure 11. Tessellations in nature: (a) Rivers forming treelike figures in the Desert of Baja California, Mexico13; (b) Forming of ice crystals14; (c) Closeup photo of Romanesco Broccoli15. 13 Taken from https://www.nationalgeographic.com/photography/photo-of-the-day/2012/4/baja-california-rivers/ 14 Taken from https://media.mnn.com/assets/images/2013/04/fractal-ice.jpg 15 Taken from https://media.mnn.com/assets/images/2013/04/broccoli-1.jpg 9  MATH 1100 Department of Mathematics and Physics Answers to Self-Assessment Questions (SAQ) ASAQ1. Their respective lines of symmetry are colored red. ASAQ2. (a) Starfish: Yes, it has reflectional symmetries. There are 5 lines of reflectional symmetries. (b) Recycle symbol: no reflectional symmetry. No lines of symmetry. ASAQ3. (a) Flower with four petals: Yes, it has rotational symmetries. Its order of rotation is 4. Its smallest angle of rotation is 360°/4 = 90°. Its rotational symmetries are 90° rotational symmetry, 2(90°) = 180° rotational symmetry, 3(90°) = 270° rotational symmetry and 4(90°) = 360° = 0° rotational symmetry. (b) Starfish: Yes, it has rotational symmetries. Its order of rotation is 5. Its smallest angle of rotation is 360°/5 = 72°. Its rotational symmetries are 72° rotational symmetry, 2(72°) = 144° rotational symmetry, 3(72°) = 216° rotational symmetry, 4(72°) = 288° rotational symmetry and 5(72°) = 360° = 0° rotational symmetry. ASAQ4. Leaves, footprints, car tire prints. ASAQ5. Little florets in the head of a sunflower (with Fibonacci sequence spirals), shells of snails, pine cone (with Fibonacci sequence spirals). ASAQ6. Tessellations sometimes exhibit reflectional, rotational or translation symmetry. However, these symmetries are not always present in a tessellation. Sometime, a tessellation does not have any symmetry at all. ASAQ7. The method to create this curve is to start with a single line segment. Divide the line segment into four equal parts, remove the second and third part and replace it with the three sides of a square (above the line for the second part and below the line for the third part) of length equal to the length of the segment that has been removed. Then repeat the process to each of the resulting line segments. 10  MATH 1100 Department of Mathematics and Physics 2. NUMBERS IN NATURE Patterns in nature can be linked to some interesting known numbers or series of numbers. 2.1 Fibonacci Sequence Fibonacci created a problem that concerns the birth rate of rabbits. The problem is stated as follows: At the beginning of a month, you are given a pair of newborn rabbits. After a month the rabbits have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbit dies, how many pairs of rabbits will there be at the start of each succeeding month? The solution to the problem created by Fibonacci is a sequence of numbers called Fibonacci sequence. Fibonacci then discovered that the number of pairs of rabbits for any month after the first two months can be determined by adding the numbers of pairs of rabbits in each of the two previous months. Figure 11 shows the number of pairs of rabbits after the first 6 months. The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … is called Fibonacci sequence. The numbers in the sequence are called Fibonacci numbers. Figure 11. Number of pairs of rabbits of the first 6 months. 16 16 Aufmann, R.N, et.al. (2008). Mathematical Excursions Third Edition. pp.19-20 11  MATH 1100 Department of Mathematics and Physics Example 9. Fibonacci numbers can be observed in some patterns on sunflowers. The little florets on the sunflower head has spirals (counterclockwise and clockwise). Some sunflowers have 21 and 34 spirals; some have 55 and 89 or 89 and 144 depending on the species. These pair of number of spirals forms two consecutive numbers of the Fibonacci sequence. Figure 12 shows sunflower heads with 34 counterclockwise spirals and 55 clockwise spirals. (a) (b) Figure 12. Sunflower Head Pattern with (a) counterclockwise spirals17 and (b) clockwise spirals18. However, this pattern is not true for all sunflowers. Using 657 sunflowers, Swinton et al. (2016) found out that one in five flowers did not conform to the Fibonacci sequence. Example 10. We can also find Fibonacci numbers in nature is in the number of petals different flowers have. Some lilies and iris have three petals, gumamela and calachuchi have five, some variety of sampaguita have eight, corn marigolds have 11, and some daisies have 34, 55, or even 89 petals. Fibonacci numbers also appear in the arrangement of leaves and branches in some plants. SAQ8. Can you give other patterns in nature where we can find Fibonacci numbers? 2.2 Fibonacci Numbers and Golden Ratio Golden ratio (also known as Divine Proportion) exists when a line is divided into two parts and the ratio of the longer part “𝑎” to shorter part “𝑏” is equal to the ratio of the sum “𝑎 + 𝑏” to “𝑎”. How to solve for the golden ratio? Denote the golden ratio by 𝜙. From the definition we have 17 Taken from https://momath.org/home/fibonacci-numbers-of-sunflower-seed-spirals/spiralsred 18 Taken from https://momath.org/home/fibonacci-numbers-of-sunflower-seed-spirals/spiralsgreen/ 12  MATH 1100 Department of Mathematics and Physics 𝑎 𝑎+𝑏 = = 𝜙. 𝑏 𝑎 We have 𝑎 𝑎+𝑏 𝜙= and 𝜙 =. 𝑏 𝑎 𝑎 1 𝑏 From 𝜙 = 𝑏 , we get = 𝑎. 𝜙 𝑎+𝑏 Now we use 𝜙 =. 𝑎 𝑎 𝑏 𝜙= + 𝑎 𝑎 𝑏 𝜙 = 1+ 𝑎 1 𝜙 = 1+ 𝜙 Multiply both sides by 𝜙 𝜙2 = 𝜙 + 1 𝜙2 − 𝜙 − 1 = 0 Solving this quadratic equation will give us 1 + √5 𝜙= = 1.6180339887 ⋯ 2 The value of the Golden Ratio is given by the irrational number 𝜙 = 1.6180339887 …. Any two successive numbers in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, … have a ratio very close to the golden ratio. To show a few, consider the following. 5 = 1.6667 3 8 = 1.6000 5 13 = 1.6250 8 21 = 1.6154 13 If you continue, the ratio will get closer and closer to 1.618. SAQ9. Will you search which other parts in nature can we find golden ratio? 13  MATH 1100 Department of Mathematics and Physics 2.3 Creating A Fibonacci Spiral Do you know that spiral shapes also exhibit the Fibonacci sequence? Here is where the Fibonacci numbers come in. Starting with two 1 x 1 squares next to each other, draw a 2 x 2 square on top (or below) of the two 1 x 1 squares to produce a 3 x 2 rectangle. Then we draw a 3 x 3 square next to the 3 x 2 rectangle to produce a 5 x 3 rectangle. Next we draw a 5 x 5 square to produce an 8 x 5 rectangle. Just continue adding squares and you will get sets of rectangles (also called as golden rectangles since the length to width ratio is 𝜙 = 1.618 ⋯) whose sides are two successive Fibonacci numbers in length and squares with sides which are Fibonacci numbers (Figure 13). If we draw curves through the diagonal of each square, we create a spiral-like shape known as Fibonacci spiral. Figure 13. Fibonacci Rectangle19 The Fibonacci spiral on one hand is a good approximation of spirals that are present in nature such as the nautilus shell (Figure 14a), hurricanes (Figure 14b), and the human ear (Figure 14c). (a) (b) (c) Figure 14. Spirals that are Present in Nature20 19 Taken from http://i.stack.imgur.com/eHWK9.png 20 Taken from http://www.momtastic.com/webecoist/2012/10/29/the-golden-spiral-complex-geometries-in-nature/ 14  MATH 1100 Department of Mathematics and Physics 2.4 The Number 𝒆 Approximately equal to 2.718281828459045 ⋯, the irrational number 𝑒 is often referred to as Euler’s (pronounced “Oiler”) number after the Swiss mathematician Leonhard Euler who introduced the letter 𝑒 for the constant. Euler also discovered many of its remarkable properties including its being an irrational number. The number 𝑒 is also referred to as Napier’s constant after John Napier who introduced it earlier in a table of appendix for his work on logarithms. However, its discovery is attributed to Jacob Bernoulli (not Euler nor Napier), when he tried to solve a problem related to continuous compound interests. 2.4.1 Compound Interest Interest is a payment charged for borrowing a money or an income for keeping a money in a bank or making an investment. There are two ways to compute an interest: simple and compound. Simple interest is interest paid on the principal only. For example you borrowed ₱1,000 at 5% annual interest rate for 2 years. If simple interest is applied, you would pay an interest of ₱50 each year for two years. The amount of interest will not change as long as no additional money is borrowed. Compound interest on the other hand, is the addition of interest to the original principal. In other words, the interest earned also earns interest.We see from our previous discussion that money earning interest compounded continuously grows exponentially. Thus, if compounding is continuous, the accumulated balance at the end of a compounding period is given by 𝐴 = 𝑃𝑒 𝑟𝑡 (1) where 𝐴 = accumulated balance after a time 𝑡 𝑃 = principal amount 𝑟 = interest rate in decimal 𝑡 = time in years 𝑒 = 2.718 (approximately) Example 11. Mary opened a savings account with ₱10,000.00 initial deposit. If the account earns 8% interest, compounded continuously, how much would be her money after 3 years? How much would be her money after 3 years if simple interest is applied? Solution: It is given that 𝑃 = 10,000 pesos 𝑟 = 0.08 𝑡 = 3 years 15  MATH 1100 Department of Mathematics and Physics (a) Substituting these values into (1), we have 𝐴 = 10,000𝑒 (0.08)3 𝐴 = 12,712.49 Hence, Mary’s money after 3 years would be ₱12,712.49. SAQ10 Answer the second question. Example 12. How long will it take for ₱2,000 to double if it is deposited in a bank that pays 3.5% interest rate compounded continuously? Solution: It is given that 𝑃 = 2,000 pesos 𝑟 = 0.035 𝐴 = 4,000 pesos Substituting these values into (1), we have 4,000 = 2,000𝑒 0.035𝑡 Divide both sides of the equation by 2,000 4,000 2,000𝑒 0.035𝑡 = 2,000 2,000 2 = 𝑒 0.035𝑡 0.035𝑡 =2 𝑒 Take the natural logarithm of both sides ln𝑒 0.035𝑡 = ln 2 0.035𝑡 = ln 2 Divide both sides of the equation by 0.035 0.035𝑡 ln 2 = 0.035 0.035 𝑡 = 19.8 years Hence, it will take 19.8 years for the intial deposit of Php4,000 to double at the given interest rate. SAQ11 Robert received a certain amount from his parents as graduation gift. Instead of spending it, he opened an account that earns 3.5% interest compounded continuously. After 4 years, his account contains ₱23,005.48. How much did Robert receive from his parents as graduation gift? 2.4.2 Population Growth Mathematics also play a vital role in modeling the growth of population. Specifically, the exponential and logarithmic functions are applied to describe the relationship between time and population size. Different models had been formulated to project population 16  MATH 1100 Department of Mathematics and Physics growth and one of these is the Malthusian growth model or simple exponential growth model. It is named after Thomas Robert Malthus. The Malthusian model is applied in obtaining population growth of bacteria and even of humans on the assumption that resources are unlimited and the population has a continuous birth rate throughout time. This model has the following form: 𝑃(𝑡) = 𝑃𝑜 𝑒 𝑟𝑡 (2) where 𝑃(𝑡) = the population after time 𝑡 𝑃𝑜 = the initial population 𝑟 = the population growth rate in decimals 𝑡 = time 𝑒 = 2.718 (approximately) Example 13. According to United Nation estimates, the total population in the Philippines for the year 2018 is 106. 51 million, the 13 th largest in the world (Philippines Population, 2018). Census data shows that the population growth rate is 1.52%. Using the Malthusian model, project the population of the Philippines 5 years after. Solution: It is given that 𝑃𝑜 = 106.51 million 𝑟 = 0.0152 𝑡 = 5 years Substituting these values into (1), we have 𝑃 (5) = 106.51𝑒 0.0152(5) = 114.92 million Hence, there will be approximately 114.92 million people in the Philippines by 2023. However, in real life, the population does not grow exponentially forever. If that happens, the population of humans, animals and bacteria will become very very large to the point that the population will outgrow the planet earth. In reality, the growth rate slows down due to many factors such as diseases, calamity, limited resources, etc. For this reason, Pierre Verhulst proposed in 1836 an alternate model that allows for a fact that there are constraints in population growth. The model is known as logistic growth model and is of the form: 𝐾 𝑃 (𝑡 ) = (3) 1+𝐴𝑒 −𝑘𝑡 where 𝑃 (𝑡) = the population after time 𝑡 𝐾 = carrying capacity or limiting value 17  MATH 1100 Department of Mathematics and Physics 𝑘 = relative growth rate coefficient 𝐾−𝑃 𝐴= 𝑃 0 0 𝑃0 = the initial population at time 𝑡 = 0 Example 14. The population of a certain species of fish is modeled by a logistic growth model with relative growth rate of 𝑘 = 0.3. One hundred fish are initially introduced into the pond with maximum carrying capacity of 500. Assuming that fish are not harvested, (a) estimate the number of fish in the pond after one year; (b) estimate the time it will take for there to be 350 fish in the pond. Solution: Since it is given that 𝑃0 = 100, the initial population 𝑘 = 0.3 𝐾 = 500 , the maximum carrying capacity of the pond then, 𝐾 − 𝑃0 𝐴 = 𝑃0 𝐴 500 − 100 = 100 𝐴 =4 𝐾 Substituting these values in Formula (3), 𝑃(𝑡) = 1+𝐴𝑒−𝑘𝑡 , we have 500 (a) 𝑃 (1) = = 126.15 fish 1+4𝑒 −0.3(1) Since the actual number of fish must be a whole number, we round it off to 126. That is, there will be 126 fish in the pond after one year. (b) Since it is given that 𝑃(𝑡) = 350 which is the number of fish in the pond after time 𝑡, then 500 350 = 1 + 4𝑒 −0.3(𝑡) 500 1 + 4𝑒 −0.3𝑡 = 350 10 1 + 4𝑒 −0.3𝑡 = 7 10 4𝑒 −0.3𝑡 = −1 7 3 4𝑒 −0.3𝑡 = 7 18  MATH 1100 Department of Mathematics and Physics 4𝑒 −0.3𝑡 3 4 =7 4 3 𝑒 −0.3𝑡 = 28 Take the natural logarithm of both sides 3 ln 𝑒 −0.3𝑡 = ln 28 3 −0.3𝑡 = ln ( ) 28 1 3 𝑡 = − 0.3 ln (28) = 7.45 Hence, it will take 7.45 years for the pond to have 350 fish. SAQ12 Influenza B virus can be spread by direct transmission such as coughing, sneezing or spitting. Suppose there are two pupils in a class of 40 children who was infected by the virus. Assuming none of the children has flu vaccine before, estimate (Let the logistic growth constant 𝑘 be equal to 0.6030). (a) the number of children who will catch the virus after 3 days. (b) estimate the time it will take for 20 children to catch the virus. 2.4.3 Exponential Decay From the previous section we learned that a quantity exhibits exponential growth if it increases continuously according to the model 𝑃 (𝑡) = 𝑃𝑜 𝑒 𝑟𝑡. On the other hand, if the quantity decreases continuously at a rate 𝑟, 𝑟 > 0, then we have an exponential decay and it is modeled by the function 𝑃 (𝑡) = 𝑃𝑜 𝑒 −𝑟𝑡 (4) where 𝑃 (𝑡) = the quantity at any time 𝑡 𝑃𝑜 = the initial quantity 𝑟 = rate of decay in decimals 𝑡 = time 𝑒 = 2.718 (approximately) Notice that the models for exponential growth and exponential decay are of the same form except for the negative sign in the exponent. Examples of exponential decay are radioactive decay, radiocarbon dating, drug concentration in the blood stream and depreciation value. 19  MATH 1100 Department of Mathematics and Physics Example 15. A certain radioactive element has an annual decay rate of 12%. If there is a 100-gram sample of the element right now, how many grams will be left in 3 years? What is the half-life of the said radioactive element? Solution: (a) It is given that 𝑃0 = 100 grams 𝑟 = 0.12 𝑡 = 3 years Substituting these values into Formula (4), we have 𝑃 (3) = 100𝑒 (−0.12)(3) = 69.77 grams Hence after 3 years, there will only remain 69.77 grams. (b) We want to find the time required for the amount of the radioactive element to reduce to half its initial value. That is, we want to solve for 𝑡 when 𝑃(𝑡) = 50 grams. Substituting this value into Formula (4), we have 50 = 100𝑒 (−0.12)𝑡 50 100𝑒 (−0.12)𝑡 = 100 100 0.5 = 𝑒 (−0.12)𝑡 ln 0.5 = ln 𝑒 (−0.12)𝑡 ln 0.5 = (−0.12)𝑡 ln 𝑒 ln 0.5 = (−0.12)𝑡 ln 0.5 (−0.12)𝑡 = −0.12 −0.12 ln 0.5 𝑡 =− 0.12 𝑡 = 5.8 years Thus, the amount of radioactive element is reduced by 50% every 5.8 years. SAQ13 Manny takes 500 mg of ibuprofen to relieve pain from arthritis. Each hour, the amount of ibuprofen in his system decreases by 25%. How much ibuprofen is left after 4 hours? 20  MATH 1100 Department of Mathematics and Physics Answers to SAQ SAQ10 Mary opened a savings account with ₱10,000.00 initial deposit. If the account earns 8% interest, how much would be her money after 3 years if simple interest is applied? Solution: If simple interest is applied, the account of Mary would earn ₱10,000.00(0.08) = ₱800 each year. In 3 years, she would earn ₱800(3) = ₱2,400. So her account would have a total of ₱10,000 + ₱2,400 = ₱12,400. SAQ11 Robert received a certain amount from his parents as graduation gift. Instead of spending it, he opened an account that earns 3.5% interest compounded continuously. After 4 years, his account contains ₱23,005.48. How much did Robert receive from his parents as graduation gift? Solution: 23,005.48 = (𝑃)𝑒 0.035 (4) 23,005.48 =𝑃 𝑒 0.035 (4) 𝑃 = 20,000 Robert received ₱20,000.00 from his parents as graduation gift. SAQ12 Influenza B virus can be spread by direct transmission such as coughing, sneezing or spitting. Suppose there are two pupils in a class of 40 children who was infected by the virus. Assuming none of the children has flu vaccine before, estimate (Let the logistic growth constant 𝑘 be equal to 0.6030). (a) the number of children who will catch the virus after 3 days. (b) estimate the time it will take for 20 children to catch the virus. Solution: 𝑃(3) = the population after 3 days 𝐾 = carrying capacity or limiting value = 40 𝑘 = relative growth rate coefficient = 0.6030 𝐾−𝑃 𝐴= 𝑃 0 0 𝑃0 = the initial population at 0 days =2 21  MATH 1100 Department of Mathematics and Physics (a) 𝐾 − 𝑃0 𝐴 = 𝑃0 40 − 2 𝐴 = = 19 2 𝐾 𝑃(𝑡) = 1 + 𝐴𝑒 −𝑘𝑡 40 𝑃 (3) = 1 + 19𝑒 −0.6030(3) 𝑃(3) = 9.7264 𝑃 (3) ≈ 10 children (b) 𝐾 𝑃(𝑡) = 1 + 𝐴𝑒 −𝑘𝑡 40 20 = 1 + 19𝑒 −0.6030(𝑡) 40 1 + 19𝑒 −0.6030(𝑡) = 20 −0.6030(𝑡) = 2 1 + 19𝑒 19𝑒 −0.6030(𝑡) = 2 − 1 19𝑒 −0.6030(𝑡) = 1 19𝑒 −0.6030(𝑡) 1 = 19 19 1 𝑒 −0.6030(𝑡) = 19 Take the natural logarithm of both sides 1 ln(𝑒 −0.6030(𝑡) ) = ln ( ) 19 1 −0.6030(𝑡) ln 𝑒 = ln ( ) 19 1 −0.6030(𝑡) = ln ( ) 19 −0.6030𝑡 1 ln (19) −0.6030 = −0.6030 𝑡 = 4.8830 days 𝑡 ≈ 5 days In 5 days, 20 children will catch the virus. 22  MATH 1100 Department of Mathematics and Physics SAQ13 Manny takes 500 mg of ibuprofen to relieve pain from arthritis. Each hour, the amount of ibuprofen in his system decreases by 25%. How much ibuprofen is left after 4 hours? Solution: 𝑃0 = 500 mg 𝑟 = 0.25 𝑡 = 4 hrs 𝑃 (𝑡 ) 𝑃0 𝑒 −𝑟𝑡 𝑃(4) = 500𝑒 (−0.25)(4) 𝑃(4) = 100𝑒 −1 𝑃(4) = 183.93 mg After 4 hours there will only be 183.93 mg of ibuprofen in his body. SUMMARY  A symmetry of an object in the plane is a rigid motion of the plane that leaves the object unchanged.  Reflection symmetry also called bilateral or mirror symmetry is a flipping motion.  Rotational or radial symmetry is a rigid motion that makes an object look exactly the same as it did before it was rotated about a fixed point, called the center.  Translations are transformations that slide objects along without rotating them (Stewart, 1995). A translation also preserves orientation.  A spiral is formed because of a property of growth known as self-similarity or scaling, which means that the same shape is maintained (not of the same size) as the object (creature) grows.  Tessellation (or tiling) is a pattern made up of one or more geometric shapes that are joined together without overlaps or gaps to cover a plane.  A fractal is a never ending replication of a pattern at different scales (same shape but different size). This property is called self-similarity.  The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … is called Fibonacci sequence. The numbers in the sequence are called Fibonacci numbers.  Golden ratio (also known as Divine Proportion) exists when a line is divided into two parts and the ratio of the longer part “𝑎” to shorter part “𝑏” is equal to the ratio of the sum “𝑎 + 𝑏” to “𝑎”. The value of the Golden Ratio is given by the irrational number 𝜙 = 1.6180339887 …  The Euler number 𝑒 which is an irrational number is approximately equal to 2.718281828459045 ⋯. It is used in the application of compound interest, population growth and exponential decay. 23  MATH 1100 Department of Mathematics and Physics POST-ASSESSMENT To evaluate your understanding of the lessons, kindly answer the following problems below. I. Essay/Application. Take a selfie with an object that has a symmetry on it. This object must be located inside your house or your backyard. Identify and discuss the symmetry that is present in the object. II. Problem Solving. Show the solution to the following problems below. 1. Peter deposited ₱4,000.00 into an account which earns 10% interest, compounded continuously. How much would be his money after 5 years? How much would be his money after 5 years if simple interest is applied? 2. A scientist started with a culture of 40 bacteria in a dish. The number of bacteria at the end of each successive hour increased exponentially, so that the number at the end of one day was 440. To the nearest million, how many bacteria were there after one week? REFERENCES Azad, Kalid (n.d.). An Intuitive Guide To Exponential Functions & e. Retrieved 12/27/2018 from https://betterexplained.com/articles/an-intuitive-guide-to- exponential-functions-e/ Dall sheep. Wikipedia. Retrieved 6/18/2018 from https://en.wikipedia.org/wiki/Dall_sheep Golden Spiral. Wikipedia. Retrieved 6/14/2018 from https://en.wikipedia.org/wiki/Golden_spiral e (mathematical constant). Wikipedia. Retrieved 12/27/2018 from https://en.wikipedia.org/wiki/E_(mathematical_constant) Hutchinson, J. (2010). An Introduction to contemporary mathematics (online). Retrieved 6/11/2018 from https://maths.anu.edu.au/files/introduction_contemporary_mathematics.pdf Jonathan Swinton, Erinma Ochu, & The MSI Turing’s Sunflower Consortium (2016). Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment. Royal Society Open Science. Retrieved 6/14/14 from http://rsos.royalsocietypublishing.org/content/3/5/160091. 24  MATH 1100 Department of Mathematics and Physics Knott, R. (2016). Fibonacci numbers and nature (online). Retrieved 6/11/2018 from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html Lipkin, Leonard & Smith, David (2001). Logistic Growth Model. Journal of Online Mathematics and its Applications. Retrieved June 24, 2018 from https://www.maa.org/book/export/html/115630 Murali, Sruthy (2012). Golden Ratio in Human Anatomy. Retrieved 12/18/2018 from https://www.researchgate.net/publication/234054763_GOLDEN_RATIO_IN_HUMA N_ANATOMY National Museum of Mathematics. The coolest thing that ever happened to math! Retrieved 6/13/2018 from https://momath.org/home/fibonacci-numbers-of- sunflower-seed-spirals/ Philippines Population (2018). Retrieved June 22, 2018 from http://worldpopulationreview.com/countries/philippines-population/ Rigid Transformations – Isometries. MathBitsNotebook Geometry. Retrieved January 7, 2019 from https://mathbitsnotebook.com/Geometry/Transformations/TRRigidTransformations.html Stewart, Ian (1995). Nature’s numbers: the unreal reality of mathematics. Basicbooks, A Division of HarperCollins Publishers, Inc. The Golden Spiral: Complex Geometries in Nature. Retrieved 6/14/2018 from http://www.momtastic.com/webecoist/2012/10/29/the-golden-spiral-complex- geometries-in-nature/ 25