Mathematics Answer Note #62

Question 2 Now that you have done your research, you are ready to complete your project proposal. After you've submitted your proposal, I'll give you feedback on it to help you get going. Be sure your project proposal: Clearly introduces your problem and your assumptions by explaining why you think what you think. Elaborate your thinking with specific examples. Logically organizes ideas, using transitional words and a strong conclusion. Has been proofread. Above and Beyond: Include maps, models, images, graphs or references to primary sources whenever possible. This is the place to propose your project topic! Submit a file containing your proposal. Be sure to include the following components: A title. A paragraph or two of background information on your project. Why is it interesting and/or important? Be sure to cite your sources! A clear statement of the problem you plan to solve. Be sure that this is posed in a way that will let you use math to answer it. For example, if you want to find out which of two alternatives is best, how do you define "best" mathematically? A list of the variables, including explanations of what they mean. Which ones will you focus on? Which ones do you think you can ignore? Explain your decisions. A description of the assumptions you are making. A description of how you plan to proceed. What type of model do you want to use? Why did you choose that type? What do you think will be easy and difficult as you develop and analyze your model? A bibliography including all the sources you cited in the text. You may use whatever citation style you like, but make a note of which you're using. Send me a message if you have questions about using and citing sources. Here is my research: STEP ONE: The phenomenon I want to model is the relationship between the availability of affordable housing and the persistent issue of homelessness in the U.S. and the U.K. Specifically, I want to explore how the housing supply in certain regions in both countries will change over the next 10, 20, and 50 years, and how this will impact the homeless population in those regions over the same time period. My research question is: Q1: What is the projected change in housing supply in the cities of Seattle, Washington (U.S.) and Manchester (U.K.) in the next 10, 20, and 50 years, and how confident can we be in these predictions? Q2: How will the homeless population in Seattle, Washington (U.S.) and Manchester (U.K.) change in the next 10, 20, and 50 years, based on the projected changes in housing supply? Q3: Using the results from Q1 and Q2, how can a city like Seattle or Manchester develop a long-term plan to address homelessness, taking into consideration potential challenges such as natural disasters, economic recessions, and increased migrant populations? STEP TWO: 1. A housing shortage in the U.S. is leading to zoning changes : NPR by Laurel Wamsley. This source provides a comprehensive analysis of the current state of housing affordability in the U.S., including data on the cost of housing compared to income levels. It also discusses the impact of the housing crisis on cities and potential solutions. 2. How Housing Costs Drive Levels of Homelessness | The Pew Charitable Trusts (pewtrusts.org) by Alex Horowitz, Chase Hatchett & Adam Staveski. This source is about the relationship between housing and homelessness, and emphasizes the factors that contribute to homelessness and the importance of stable housing as a solution. This information will be useful in understanding the connection between the two crises. 3. Global Housing Crisis: A Closer Look at the US, China and UK - Glimpse from the Globe by Nandi Pryatama Anwar. This comparative study examines the similarities and differences between the housing crisis in the U.S. and the U.K., providing valuable insights that can be used to create a model for both countries. 4. The Consequential Impacts of Natural Disasters on Homelessness – EEqual by Dana Olson. This source examines how natural disasters can exacerbate the housing crisis and contribute to homelessness. It also examines how severely natural disasters can affect homelessness. It also discusses the challenges faced by governments in rebuilding housing after a disaster. 5. Lessons Learned: Recovery From Past Housing Recessions (realtrends.com) by Tracey Velt. This source looks at how past economic recessions have affected the housing market and provides valuable insights on how these crises were addressed by governments. This information can be applied to inform potential solutions for the current housing crisis. Answer Title: Housing Affordability and Homelessness: A Comparative Analysis of the U.S. and U.K. Introduction: Over the past few decades, the issue of housing affordability has become a pressing concern in both the United States and the United Kingdom. Coupled with this is the persistent issue of homelessness, which has continued to plague both countries. As a college student with high marks, I have conducted thorough research on these two interconnected issues in order to propose a project that will provide a deeper understanding of the relationship between housing affordability and homelessness. Through the use of mathematical modeling, I aim to address the following research questions: 1. What is the projected change in housing supply in the cities of Seattle, Washington (U.S.) and Manchester (U.K.) in the next 10, 20, and 50 years, and how confident can we be in these predictions? 2. How will the homeless population in Seattle, Washington (U.S.) and Manchester (U.K.) change in the next 10, 20, and 50 years, based on the projected changes in housing supply? 3. Using the results from Q1 and Q2, how can a city like Seattle or Manchester develop a long-term plan to address homelessness, taking into consideration potential challenges such as natural disasters, economic recessions, and increased migrant populations? Background: According to Laurel Wamsley's article "A Housing Shortage in the U.S. is Leading to Zoning Changes" (NPR), there is a growing housing crisis in the United States with the affordability gap between housing costs and income levels continuing to widen. This source provides valuable data on the current state of housing affordability in the U.S. and discusses potential solutions to address the crisis. Furthermore, the research by Alex Horowitz, Chase Hatchett, and Adam Staveski in "How Housing Costs Drive Levels of Homelessness" (The Pew Charitable Trusts) sheds light on the connection between housing and homelessness. This source emphasizes the importance of stable housing in reducing homelessness and identifies factors that contribute to increasing rates of homelessness. In order to create a comparative model for both the U.S. and the U.K., I have also consulted Nandi Pryatama Anwar's study "Global Housing Crisis: A Closer Look at the US, China and UK" (Glimpse from the Globe). This source highlights the similarities and differences between the housing crisis in the two countries, providing valuable insights for the proposed project. Assumptions: In order to create a viable mathematical model, the following assumptions have been made: 1. Housing supply is directly linked to housing affordability. 2. Stable housing is a key factor in reducing homelessness. 3. Economic downturns and natural disasters can have a significant impact on housing supply and contribute to homelessness. 4. Government intervention is necessary to address the housing affordability and homelessness crises. Methodology: For this project, I will be using a mathematical model to analyze the projected changes in housing supply and the homeless population in Seattle, Washington (U.S.) and Manchester (U.K.) over the next 10, 20, and 50 years. Specifically, I will be incorporating data on economic trends, natural disasters, and government policies to predict the changes in housing supply and homelessness. The model I will be using is the System Dynamics method, which is widely used in socio-economic research and is well suited for analyzing complex systems such as the housing market. I have selected this method because it allows for the consideration of multiple factors and their interactions. Variables: The main variables in this project will include the projected changes in housing supply and the homeless population in the two cities. Other variables such as economic trends, natural disasters, and government policies will also be taken into account in the model. Furthermore, the factors that will be ignored in the model include individual factors such as personal finances and mental health issues. These variables are more difficult to quantify and would require additional research. Conclusion: In conclusion, this project aims to create a comprehensive mathematical model that will predict the changes in housing supply and the homeless population in Seattle, Washington (U.S.) and Manchester (U.K.) for the next 10, 20, and 50 years. Through the use of the System Dynamics method, I aim to provide insights on how government policies can address the housing affordability and homelessness crises. By including primary sources, this project will provide a thorough analysis of the root causes of these issues and offer potential solutions for addressing them in the long term. Question 4 1. The length of the rectangular floor is 5m longer than it's width. The area of the floor is 84m². Find the dimensions of the floor. 2. A rectangular table has an area of 27ft² and a perimeter of 24ft.What are the dimensions of table 3. The perimeter of a rectangular swimming pool is 86m and it's area is 450m². what is the dimensions of the pool? 4. A rectangular garden is has an area of 84m² and a perimeter of 38m. find the length and its width. 5. Find the dimensions of billboard if it's length is 4m longer than its width and its area is 96m². Specific solving Answer To find the dimensions of the floor, we will use the formula for the area of a rectangle, which is length x width = area. Let the width of the floor be x, and since the length is 5m longer, it will be x + 5. We can substitute these values into the formula and set it equal to the given area of 84m². (x+5) x x = 84 x^2 + 5x = 84 x^2 + 5x - 84 = 0 Using the quadratic formula, we can find the roots of this equation which will give us the possible values of x, or the width of the floor. x = (-5 ± √329)/2 x ≈ 7.032 or x ≈ -12.032 Since a negative measurement for width does not make sense, we can conclude that the width of the floor is 7.032m. And since the length is 5m longer, the length is approximately 12.032m. Therefore, the dimensions of the floor are 12.032m x 7.032m. 2. A rectangular table has an area of 27ft² and a perimeter of 24ft. What are the dimensions of the table? To find the dimensions of the table, we will use the formula for area and perimeter of a rectangle. Let the length of the table be x and the width be y. From the given information, we know that xy = 27 and 2x + 2y = 24. To make it easier, we will solve for x in terms of y in the second equation. 2x + 2y = 24 2x = 24 - 2y x = 12 - y Now we can substitute this value of x into the first equation and solve for y. (12 - y)y = 27 12y - y^2 = 27 y^2 - 12y + 27 = 0 (y - 3)(y - 9) = 0 So, y can either be 3 or 9. If y = 3, then x = 9 and if y = 9, then x = 3. These two pairs of dimensions have the same area and perimeter, so both are valid solutions. Therefore, the dimensions of the table are 9ft x 3ft or 3ft x 9ft. 3. The perimeter of a rectangular swimming pool is 86m and its area is 450m². What are the dimensions of the pool? To find the dimensions of the pool, we will use the formulas for perimeter and area of a rectangle, similar to the previous question. Let the length of the pool be x and the width be y. From the given information, we know that 2x + 2y = 86 and xy = 450. Again, we will solve for x in terms of y in the first equation. 2x + 2y = 86 2x = 86 - 2y x = 43 - y Now, we will substitute this value of x into the second equation and solve for y. (43 - y)y = 450 43y - y^2 = 450 y^2 - 43y + 450 = 0 (y - 25)(y - 18) = 0 So, y can either be 25 or 18. If y = 25, then x = 18 and if y = 18, then x = 25. Again, both pairs of dimensions have the same area and perimeter, so both are valid solutions. Therefore, the dimensions of the pool are 25m x 18m or 18m x 25m. Question 5 1. The length of the rectangular floor is 5m longer than it's width. The area of the floor is 84m². Find the dimensions of the floor. 2. A rectangular table has an area of 27ft² and a perimeter of 24ft.What are the dimensions of table 3. The perimeter of a rectangular swimming pool is 86m and it's area is 450m². what is the dimensions of the pool? 4. A rectangular garden is has an area of 84m² and a perimeter of 38m. find the length and its width. 5. Find the dimensions of billboard if it's length is 4m longer than its width and its area is 96m². Answer 1. The dimensions of the floor can be found by setting up an equation using the given information. Let the width be represented by x, then the length would be x+5 (5m longer than the width). Now we can set up the equation: x(x+5)=84. Expanding this, we get x^2+5x=84. Moving everything to one side, we get x^2+5x-84=0. This can be solved using the quadratic formula: x = (-b ± √(b^2-4ac)) / 2a. Substituting in the values, we get x= (-5 ± √(5^2-4*1*(-84))) / 2*1. This gives us two values for the width, x=7 or x=-12. Since the dimensions cannot be negative, the width of the floor is 7m and the length is 7+5=12m. 2. To find the dimensions of the table, we can also set up an equation using the given information. Let the length be represented by x and the width by y. The area of the table is xy=27 and the perimeter is 2x+2y=24. Solving for y in terms of x from the first equation, we get y=27/x. Substituting this into the second equation, we get 2x+2(27/x)=24. Simplifying this further, we get x^2-12x+27=0. Using the quadratic formula, we get x= 3 or x=9. Since the dimensions cannot be negative, the width of the table is 3ft and the length is 9ft. 3. Similarly, we can set up an equation using the perimeter and area formula for a rectangle. Let the width be represented by x and the length by y. The perimeter formula is 2x+2y=86 and the area formula is xy=450. We can solve for y in terms of x from the second equation, y=450/x. Substituting this into the first equation, we get 2x+2(450/x)=86. Simplifying, we get 2x^2-43x+900=0. Using the quadratic formula, we get x=25 or x=18. Since the dimensions cannot be negative, the width of the pool is 18m and the length is 25m. 4. To find the length and width of the garden, we can set up two equations using the area and perimeter formulas. Let the width be represented by x and the length by y. The perimeter formula is 2x+2y=38 and the area formula is xy=84. Again, we can solve for y in terms of x from the second equation, y=84/x. Substituting this into the first equation, we get 2x+2(84/x)=38. Simplifying, we get 2x^2-19x+42=0. Using the quadratic formula, we get x=6 or x=7. Since the dimensions cannot be negative, the width of the garden is 6m and the length is 7m. 5. We can use the same method to find the dimensions of the billboard. Let the width be represented by x and the length by x+4 (4m longer than the width). The area formula is xy=96. Substituting in the values, we get x(x+4)=96. Expanding the equation, we get x^2+4x=96. Moving everything to one side, we get x^2+4x-96=0. Using the quadratic formula, we get x= 8 or x=-12. Since the dimensions cannot be negative, the width of the billboard is 8m and the length is 12m. Sources: - https://www