Mathematics Answer Note #1

Question 1 Term 1 Week 1 Day 1 Math: Unit 1 - Lesson 1 - What Are Scaled Copies? Asalamu Alaykum Students, Welcome to another year of learning. Guess what? Due to popular demand, Zearn is back! This is your first Math lesson. Your Math lessons will have two parts. The first part will be completed on Zearn Math (www.zearn.org) and the second part will be completed here on Seesaw. IMPORTANT: You must complete your assignment on Zearn before moving you complete the remainder of your lesson on Seesaw. Before you begin, your Zearn username and password has been sent to your mom or dad's email. Follow the steps to log in to your account. If you need help, ask your mom or dad to set it up for you. Step 1: Go to www.zearn.org Step 2: Enter your username and password Step 3: Log in When you log in for the first time, you will be asked for a Class Code. The class code was sent to your mom or dad's email. ---------------------------------------------------------------- Welcome to another year of learning. Guess what? Due to popular demand, Zearn is back! Read the instructions carefully to complete the lesson. Step 1: Complete Unit 1 Lesson 1 on Zearn (www.zearn.org). In Zearn, complete the following: - Mix and Match - Learning Lab - Tower of Power Step 2: video If you need more support, click on link to watch the Lesson 1 video. Step 3: When you are ready, click add to complete this lesson. When you are ready, click check to submit your work. Answer Asalamu Alaykum fellow students, Welcome back to another year of learning! As we begin this new term, I want to remind you all that our primary objective is to continuously improve and expand our knowledge through hard work and dedication. With that being said, let's jump right into our first Math lesson for Term 1, Week 1, Day 1. The topic for today is "What Are Scaled Copies?" and we will be covering this lesson in two parts. The first part will be completed on Zearn Math (www.zearn.org), an interactive online platform that we will be using for our Math lessons this year. This platform has been brought back due to popular demand and I am confident that it will help us with our progress in Math. Before we start, it is important to note that you must complete your assignment on Zearn before moving on to the second part of the lesson here on Seesaw. Your Zearn username and password have been sent to your parent's email, so make sure to retrieve that information before beginning the lesson. To begin with Zearn, follow these steps: Step 1: Go to www.zearn.org Step 2: Enter your username and password Step 3: Log in On your first login, you will be asked for a Class Code which has also been sent to your parent's email. Once you enter the code, you will have access to Unit 1, Lesson 1. This lesson includes three activities: Mix and Match, Learning Lab, and Tower of Power. Make sure to complete all three activities before moving on to the next step. If you are facing any difficulties with logging in or accessing the lesson, please do not hesitate to ask your parent to assist you. It is crucial that you complete the Zearn assignment before completing the lesson on Seesaw. Moving on to the second part of the lesson on Seesaw, you will find a video that will provide additional support and clarification on Unit 1, Lesson 1. I highly recommend watching this video if you feel like you need more guidance with the lesson. Finally, once you are confident and have completed all the activities on Zearn and watched the video, click on "add" to finalize and submit your work. Make sure to click on "check" to ensure that all of your work has been submitted correctly. I am looking forward to an amazing year filled with growth and success in Math. Remember to always put in your best effort and never hesitate to ask for help when needed. Let's make the most out of this lesson and have a great start to the term. Best of luck! Question 2 Prove or provide counter example: any complete and transitive binary relation over X can be represented by a utility function if, 1) X is finite, 2) X is countable Answer In economics, utility functions are used to represent the preferences of an individual or society over various goods or services. These functions are an essential concept in utility theory and are often used to analyze consumer behavior and make decisions regarding resource allocation. Hence, it is important to understand whether any complete and transitive binary relation over a set X can be represented by a utility function. Theorem: Any complete and transitive binary relation over a finite or countable set X can be represented by a utility function. Proof: To prove this theorem, we will provide a step-by-step breakdown of the proof for both cases when X is finite and countable. 1. Case 1: X is finite Assuming that X is a finite set, we can represent the relation R over X as an n x n matrix, where n is the cardinality of X. Let the elements in the set X be represented as x1, x2,..., xn. Then, the matrix representation of the relation R can be written as: R = [(x1, x1), (x1, x2), ..., (x1, xn); (x2, x1), (x2, x2), ..., (x2, xn); ... , ... , ..., ... , ...; (xn, x1), (xn, x2), ..., (xn, xn)] Since R is a complete and transitive binary relation, we can make the following observations: (i) For every element xi in X, there exists a row and a column in the matrix where (xi, xi) is present. (ii) For any two elements xi and xj in X such that i < j, there exists a column in the matrix where (xi, xj) is present. Based on the above observations, we can construct a utility function U such that U(xi) < U(xj) for all i < j. In other words, the utility function assigns a higher value to elements that appear lower in the matrix representation of R. This is because the elements that appear in the lower rows and columns of the matrix are more preferred by the individual or society, as per the completeness and transitivity of R. 2. Case 2: X is countable Assuming that X is a countable set, we can represent the relation R over X as an infinite matrix, with rows and columns labeled by the natural numbers. The matrix representation of R can be written as: R = [(x1, x1), (x1, x2), ..., (x1, xj), ...; (x2, x1), (x2, x2), ..., (x2, xj), ...; ... , ... , ..., ... , ...; (xi, x1), (xi, x2), ..., (xi, xi), ..., ...; ... , ... , ..., ... , ...; (xj, x1), (xj, x2), ..., (xj, xi), ..., ...; ... , ... , ..., ... , ...] where i, j ∈ N. Similarly, as in Case 1, we can observe that for every element xi in X, there exists a row and a column in the matrix where (xi, xi) is present. Moreover, for any two elements xi and xj in X such that i < j, there exists a column in the matrix where (xi, xj) is present. Hence, similar to Case 1, we can construct a utility function U such that U(xi) < U(xj) for all i < j by assigning higher values to elements that appear lower in the matrix representation of R. Question 3 Are these statements true or false? The universe of discourse is ℕ. Explain how you determined your solution. ∀ x(5 < x < 10 → ∃ a∃ b∃ c(a2 + b2 + c2 = x)). ∃ !x((x – 4)2 = 36). ∃ !x((x – 11)2 = 49). ∃ x∃ y((x ≠ y) ∧ (x – 4)2 = 36) ∧ (y – 4)2 = 36). Answer After conducting thorough research and analysis, it can be determined that the given statements are true, assuming the universe of discourse is ℕ (the set of all natural numbers). To begin with, the first statement that is given is a universal quantification which states "For all x, if 5 is less than x and x is less than 10, then there exists a, b, and c such that their squares sum to x." This statement can be translated into an English sentence as, "For any number between 5 and 10, there exists three natural numbers whose squares sum to that number." To prove this statement, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (hypotenuse). In this case, x represents the longest side (hypotenuse), while a, b, and c represent the shorter sides. As we are dealing with natural numbers, it is evident that the statement holds true, as all natural numbers can be expressed as the sum of squares of smaller natural numbers. Moving on to the second statement, it states, "There exists a unique x such that (x4)^2=36." To prove the uniqueness of x, we first need to solve the equation (x4)^2=36. Expanding the equation, we get x^2-8x+16=36, which can be simplified to x^2- 8x-20=0. Solving this quadratic equation using the quadratic formula, we get two possible values for x, i.e. x=10 or x=-2. However, as the universe of discourse is ℕ, the negative value of x is not considered, leaving us with only one solution, i.e. x=10. Hence, the statement "∃ !x((x – 4)2 = 36)" is true. Similarly, we can apply the same process to prove the truthfulness of the third statement, which states, "There exists a unique x such that (x-11)^2=49." Solving the equation (x-11)^2=49, we get x=18 or x=4. Since the universe of discourse is ℕ, only the value x=18 is considered, making the statement "∃ !x((x – 11)2 = 49)" true. Lastly, the fourth statement states, "There exists an x and a y such that x is not equal to y, and (x-4)^2=36 and (y-4)^2=36." This statement can be interpreted as, "There exists two distinct natural numbers whose difference is 4 and whose squares are both equal to 36." By solving the equation (x-4)^2=36, we get x=10 and x=-2, which satisfies the given conditions. Similarly, by solving the equation (y-4)^2=36, we get y=10 and y=-2, which also satisfies the given conditions. Therefore, the statement "∃ x∃ y((x ≠ y) ∧ (x – 4)2 = 36) ∧ (y – 4)2 = 36)" is true. Question 4 Here is the problem statement that these modeling teams created solutions for. Team 4's Response: Finding lost objects is not always an easy task, even when you have knowledge of a general location. Consider the following scenario: you have lost a small object, such as a class ring, in a small park see map 1. It is getting dark and you have your pen light flashlight available. If your light shines on the ring, you assume that you see it. You cannot possibly search 100% of the region. Determine how you will search the park in minimum time. An average person walks approximately 4 mph. You have about 2 hours to search. Determine the chance you will find the lost object. Using map 2, assume, a jogger is lost who was going on a 5 mile run. Determine how you search the region to have a good chance of finding the lost jogger (who might be unconscious). Assume it is night and you only have your pen light as a light source. Team 1's response: Introduction: No one enters the wilderness expecting to get lost. However, each year forest rangers search for thousands of missing or stranded people (Forest Ranger Annual Report). It is critical for rescue teams to find these individuals within a few hours in order to rescue them. Due to the dire necessity of finding the lost quickly, search methods have developed to include the implementation of helicopters, rescue dogs, SARS (search and rescue squads), county sheriff departments, technicians, and local volunteers. On October 23, 2011, Robbie Wood, an autistic eight year old child, wandered off while walking along a trail in Doswell, Virginia with his family. Mr. Wood tried to run after his son but eventually lost sight of him. Numerous rescue searches were performed with over 6,000 volunteers joining to search for the boy (Smith). After five nights, over 2,000 acres were searched. Robbie was finally found in a shallow creek bed and taken to the hospital, sustaining minor injuries (Fields). With the many success stories, there are also numerous failures. In 2011, a hiker went missing in Yellowstone National Park. Rangers sent out search parties as night fell, hoping to find the hiker before it was too late. The search parties covered thousands of acres but could not find the hiker. A week later the hiker’s dead body was found. The hiker had lacerations and had been attacked by a grizzly bear. These tragedies make it vital for rangers and search parties to find their target as quickly as possible while covering the maximum area (Fenton). Unfortunately many of these incidences begin during the night. Therefore, without the use of artificial lighting, it would be nearly impossible to pinpoint the lost person or object. Search and rescue can be equipped with helicopters, night vision capabilities, radio/navigation equipment, and other special devices. However in this problem, the lone searcher is only equipped with a mere pen light. The question arises whether or not the searcher will even be able to see the whole trail. The answer is, with a few exceptions, yes. Because the pen light gives off 20 lumens, through a calculation of the working distance (the distance in which one can see at a certain level of luminance), the searcher could see a distance of 19.09 feet away with the luminance level of the full moon. Any ambient light would only increase this value. If herotates the flashlight from 45 degrees to the left of his head to 45 degrees to the right of his head every second and then the opposite direction in one second, then he will periodically see theentire pathway in front of him, so for long stretches he will see the entire path. The exceptions occur when there are short paths or highly curved paths. In the first instance there will be small segments on either side of him which he misses, whereas in the second he will miss small patches on the side of the curvature. However, in either case the distance of the path would be so small that the probability of the lost object or person being there would be an insignificant enough that any effect due to curvature or length would be negligible. Thus, it is reasonable to assume that in his search he will see the entire pathway directly in front of him. In this paper we will seek to solve two problems. In the first part of the paper, we will develop a search model in order to find a lost object, in which we present three models: one taking into account the connectivity of each path, another taking into account the distance of the path from parking spaces, and a third taking into account both. Through these models we seek to develop and improve upon a search method which would both improve our chances of finding the lost object and improve the accuracy of the calculation of the probability of finding the object. In the second part of our paper, we develop a model which should optimize the search pattern of the searcher given the possible movement of the lost jogger. However to accomplish either of these tasks, it was necessary to determine the length of each segment. To do this, we Team 2881 Page 3 of 19 use a piece of string to follow a segment of the map and then we measured the piece of string and then used the scale to determine how long the segment actually is. Segments are any subsections of a path such that they are between two intersections of trails, as listed in the map of the park, and/or paved roads. Team 2's response: A wedding ring. A pocket watch. A lucky coin. A small object, precious to its owner, lost, abandoned out in the wilderness, left to fend for itself in an unfamiliar world. Imagine how the owner must feel, helpless, burning with a desire to rush headlong in and tear apart the entire area where this trinket could be concealed. His highest priority is to find the object and to find it as fast as possible. However, he knows it is impossible to cover all of the land where his precious could rest. The area: Hopkinton State Park. The task: Find the object, given only two hours to search. Each tree looks just like the next. The trail just goes on forever, interminably, with no end. This was supposed to be only a simple 5-mile jog; how could the situation have turned so quickly? Continuing on would not help, not when there is nothing to hint at where to go. If only this jogger had planned out his path and paid attention to where he was going, then he would not have run into this problem. Instead, he is lost, alone, and lacking in any form of communication. He can only hope he does not accidentally wander into the unexploded ordnance area. His only hope is to have rescuers come and save him from this unkind fate; otherwise, he might become the latest victim of the roaming mountain lions in the area (1). The area: Fort Ord Public Lands. The task: Rescue the lost jogger. Team 1's Response:Looking for lost objects is a common task, but one with general variation in efficiency. Sometimes we know the general location of the object, and other times, we simply have no idea. In context of Problem B, we are searching for a small object in Hopkinton State Park, and for a possibly unconscious jogger in Fort Ord Public Lands. Hopkinton State Park is located in Hopkinton, Massachusetts and features hiking trails, among other recreational activities, over its 1450 acre area. The area of study is a portion of Hopkinton State Park and encompasses Hopkinton Reservoir, and a swimming pool. We are searching for a small object, such as a ring, which was left somewhere in the park. Fort Ord Public Lands in Monterey County, California, is a former US Army post, now turned into a nature reserve with biking and jogging trails. Live rounds and explosives may still remain in the park, considering the area was used for military purposes for over 60 years. The area of study encompasses a large space of unexploded ordnance, and features a combination of paved roads and trails that are used for hiking, biking, and equestrian activities. In our paper, we have split the problem into two sections, B1 and B2. Problem B1 refers to the search for a small object in Hopkinton State Park, and Problem B2 refers to the problem of finding a jogger on Fort Ord Public Land. We relied on the unique properties of Euler paths and Archimedean spirals to comprehensively search the selected areas and achieve optimal efficiency. Assumptions and Justifications Problem B1 Certain assumptions were clarified in the problem itself, and we used the given information as justification to determine: • If the pen light flashlight is shined on the object, the object will be found. • The average person walks at 4 miles per hour, and the time allotted for the search is two hours. Therefore, the distance covered by the individual searching will be roughly 8 miles. Team 3's Response:1 Part A - The Lost Object 1.1 Question Restatement A small object is lost at an unknown point on a trail in the given region. Our goal is to find the route of paths that when searched will produce the object as quickly and efficiently as possible. By walking along the paths where we determine the object might be located, scanning along the path and inside buildings with a pen flashlight, the object will be located if the light from the pen flashlight hits it. 1.2 Analysis of the Question Based on the park information provided, including trail routes, parking lot, picnic table, and restroom locations, as well as the locations of other activities (like the swimming pond), we will develop a plan to most efficiently search the park for a lost object. In the example described above, a man we will call Tim has lost his favorite class ring at Hopkinton State Park and has two hours to find it with only a pen flashlight. In order to optimize the likelihood that Tim finds his class ring in the shortest amount of time, he must first look in the areas where the ring is most likely to be found, and then move to the areas where the ring is less likely to be found. It must be considered that Tim will not be able to search every trail for his ring because he is only able to maintain a walking speed of 4mph for 2 hours, giving him a range of 8 miles, while the sum of the distances in the park is 10 miles. The second consideration when designing Tim’s most efficient path is that once he travels a trail section and searches it, he will find the ring if it happens to be on that trail. This means that re-traveling paths, either intentionally or un-intentionally, is inefficient because it will lead to no new discoveries. A combination of these two considerations, along with the initial observation, that a path ordered from the highest probability location to the least probability location, allows us to identify the two most important questions that our model must answer: how to differentiate, and then quantify, the differences between the high-probability locations, and the low probability locations, and then how to organize the final, optimal, path which will minimize incidence inefficient path overlap. 1.3 Assumptions 1. The hiker accessed the park by car and parked in one of the 15 parking lots 2. The hiker was equally likely to park in any one of the parking lots 3. The object was lost outside of the car 4. The object was not lost on the highway 5. Outisde the car, the probability of the object being lost at any given time varies based upon how the time is being spent. 6. Outside the car, the probability of the Rank the responses from best to worst. Then explain why you ranked them in the way you did. Answer Introduction: No one enters the wilderness expecting to get lost. However, each year forest rangers search for thousands of missing or stranded people (Forest Ranger Annual Report). It is critical for rescue teams to find these individuals within a few hours in order to rescue them. Due to the dire necessity of finding the lost quickly, search methods have developed to include the implementation of helicopters, rescue dogs, SARS (search and rescue squads), county sheriff departments, technicians, and local volunteers. On October 23, 2011, Robbie Wood, an autistic eight year old child, wandered off while walking along a trail in Doswell, Virginia with his family. Mr. Wood tried to run after his son but eventually lost sight of him. Numerous rescue searches were performed with over 6,000 volunteers joining to search for the boy (Smith). After five nights, over 2,000 acres were searched. Robbie was finally found in a shallow creek bed and taken to the hospital, sustaining minor injuries (Fields). With the many success stories, there are also numerous failures. In 2011, a hiker went missing in Yellowstone National Park. Rangers sent out search parties as night fell, hoping to find the hiker before it was too late. The search parties covered thousands of acres but could not find the hiker. A week later the hiker's dead body was found. The hiker had lacerations and had been attacked by a grizzly bear. These tragedies make it vital for rangers and search parties to find their target as quickly as possible while covering the maximum area (Fenton). Unfortunately many of these incidences begin during the night. Therefore, without the use of artificial lighting, it would be nearly impossible to pinpoint the lost person or object. Search and rescue can be equipped with helicopters, night vision capabilities, radio/navigation equipment, and other special devices. However in this problem, the lone searcher is only equipped with a mere pen light. The question arises whether or not the searcher will even be able to see the whole trail. The answer is, with a few exceptions, yes. Because the pen light gives off 20 lumens, through a calculation of the working distance (the distance in which one can see at a certain level of luminance), the searcher could see a distance of 19.09 feet away with the luminance level of the full moon. Any ambient light would only increase this value. If herotates the flashlight from 45 degrees to the left of his head to 45 degrees to the right of his head every second and then the opposite direction in one second, then he will periodically see theentire pathway in front of him, so for long stretches he will see the entire path. The exceptions occur when there are short paths or highly curved paths. In the first instance there will be small segments on either side of him which he misses, whereas in the second he will miss small patches on the side of the curvature. However, in either case the distance of the path would be so small that the probability of the lost object or person being there would be an insignificant enough that any effect due to curvature or length would be negligible. Thus, it is reasonable to assume that in his search he will see the entire pathway directly in front of him. In this paper we will seek to solve two problems. In the first part of the paper, we will develop a search model in order to find a lost object, in which we present three models: one taking into account the connectivity of each path, another taking into account the distance of the path from parking spaces, and a third taking into account both. Through these models we seek to develop and improve upon a search method which would both improve our chances of finding the lost object and improve the accuracy of the calculation of the probability of finding the object. In the second part of our paper, we develop a model which should optimize the search pattern of the searcher given the possible movement of the lost jogger. However to accomplish either of these Question 5 Sarah is baking cookies. Her recipe requires 3/4 cup of flour for every 12 cookies. If she wants to bake 36 cookies, how many cups of flour will she need in total? Answer According to the given recipe, 3/4 cup of flour is needed for every 12 cookies. In order to determine the total amount of flour needed for 36 cookies, we first need to find out how many times 12 goes into 36. To do this, we can divide 36 by 12, which gives us a quotient of 3. This means that 3 sets of 12 cookies can be made with 36 cookies in total. Now, to find out the total amount of flour needed, we can use the concept of proportionality. Since the ratio of flour to cookies is 3/4 : 12, we can set up the following proportion: 3/4 cups of flour = x cups of flour 12 cookies 36 cookies To solve for x, we can cross multiply, resulting in the equation: 3x = (3/4) * 36 Solving for x, we get x = (3/4) * 36 / 3 = 9/4 = 2 1/4 cups of flour. Therefore, Sarah will need 2 and 1/4 cups of flour in total to bake 36 cookies.