Unit 2- Meadows or Malls
Report
We bring forth the best possible solution to the land allocation dilemma, after careful calculations, we have found the cheapest possible solution that will indeed satisfy both the development and recreation parties. We have for weeks studied all of the possible numbers that are a part of this intense equation, and have formulated the solution with careful consideration of mistakes. The development party will be allocated 250 acres of land, and the recreation party will be allocated 300 acres of land, for the price of approximately 353.75 grand. We are certain that this will be the most effective play since both parties will be satisfied all for the lowest possible price. The recreational party may be allocated 45% of the acreage, but for the benefit of the cost, and the sake of the efficiency of this newfounded plan, compromise must be made on their part. Though, it is quite miniscule, therefore not much arguing should be conjured up. In addition, as a simple description... in the statistics we had been given; the minimums and pre-decided terms allowed us to formulate several constraints. Afterwards, a rather simple back and forth puzzle involving a specific configuration of these constraints allowed us to come to the conclusion that this could be settled over the matter of a quite fair deal for the lowest possible price. There was a lingering thought in our minds that possibly asking the development side for the absolute minimum they would settle for, and then allocate that extra percent to the recreation party to drive down costs, though, it seemed immoral. We are certain we have found the best and most efficient way to allocate the land fairly and respectively, though that proposal is configured and formulated upon math. We wonder if there is a more efficient way that disinvolves numbers and constraints that could drive the price down lower and still satisfy the people. Both sides want as much allocated land as possible, but obviously to drive down cost for extra benefit would require us to downplay the development side as they are the more cost-demanding supporters, we believe that in lieu of obtaining information through a survey or asking of sort from the people, we must hold true to the plan we have formulated with numbers. So with that out of the way, here’s the plan. 225 of the 300 acres of ranchland will be allotted to the recreational party, and the remaining 75 will be allotted to the development party. Secondly, 75 of the 100 acres of the old army base will be assigned to the development party, and the residual 25 acres will be accredited to the recreational party. Finally, the remaining location- the Boston mine will be entirely allocated to the development party; 150 acres. Therefore, in total, the development party will receive 75 acres of ranchland, 75 acres of the army base land, and all 150 acres of the old Boston mine. In turn, the recreational party will receive 225 acres of ranchland, and 25 acres of the army base land. Now obviously we could simply ignore this and drop to 50-50, because as far down as the development party is willing to drop, the lower the cost will be. Though at this point, efficiency is a major factor as well. Shall we move forward with this proposal?
Write Up
When in the course of the endeavor to solve the Meadows or Malls problem, we came across several key fragments that built up the understanding we needed to solve this mind-boggling problem.
The corner-point principle is an essential part of this ultimate equation, and in this problem, it involves several quantities that must be determined- and are represented as variables. Those variables can then be represented using linear inequalities, allowing us to find the optimal solution that would result in the lowest cost. The problem also involved a large series of matrices- number tables that can be mathematically combined. Matrices came in handy to formulate the final constraints, as once we organized all the preset statistics, we could create constraints that we would later use to solve the problem.
The functionality of multiplying and adding matrices together can be confusing, as you have to multiply/add the first row by the columns of the second matrix, so we did end up using a calculator. One of the most important elements of the problem involving matrices was the inverse matrix, which allowed us to acquire answers from multi-step equations.
In the end, we used our 12 constraints (that we later diminished down to 9) and ordered them in certain ways to gain all the possible outcomes. Distribution of “land” for the best cost came down to a few, as many of the constraint configurations involved too large of totals- therefore being outside of the feasible region. At last, we came up with a reasonable cost, and a distribution proposal that would most likely satisfy both parties.
Reflection
Throughout the entirety of the year... I only ever gained any collaborative experience while at school. Within the period of full-online, it felt like nothing more than “here’s the facts, do this”, now I know that was never the intention, but without the physical interaction, it’s quite bland. And since we were full online for so long, it was quite impractical to work in groups, other than those assignments where we simply fill out a group document. I feel that my skills to collaborate as a whole have been kept at a pause throughout the span of online school, but being back I feel somewhat of a want to engage more immediately in groups when we have a group assignment. I believe I have gained some confidence and serenity when I participate in group discussions. I am very thankful for this and I hope it persists to make me more and more comfortable.
Full online was an attack upon my mental stability, most of it was just too easy for me to learn anything, so much was based on participation, and the few assignments that were very difficult, I could not pick up on. Thankfully, I was able to manage to pick up on at least the gradually increasing ultimate goal to solve the unit problem; I could add the few things that I did learn together to formulate an idea of what this was all about. After Spring break, when we finally came back, as many do I felt weary of the sudden change. Though, after a few days, I began to exponentially appreciate the time we were given to learn face to face, and I managed to send my confidence through the roof. And even though I did not understand a couple of key concepts, I managed to somehow collect the information I needed... as if by nothing. I’m not sure what happened, but when we came back I felt better than ever to succeed, and I am so glad that I have. I learned to strengthen my focus and use it to finish strong.
We bring forth the best possible solution to the land allocation dilemma, after careful calculations, we have found the cheapest possible solution that will indeed satisfy both the development and recreation parties. We have for weeks studied all of the possible numbers that are a part of this intense equation, and have formulated the solution with careful consideration of mistakes. The development party will be allocated 250 acres of land, and the recreation party will be allocated 300 acres of land, for the price of approximately 353.75 grand. We are certain that this will be the most effective play since both parties will be satisfied all for the lowest possible price. The recreational party may be allocated 45% of the acreage, but for the benefit of the cost, and the sake of the efficiency of this newfounded plan, compromise must be made on their part. Though, it is quite miniscule, therefore not much arguing should be conjured up. In addition, as a simple description... in the statistics we had been given; the minimums and pre-decided terms allowed us to formulate several constraints. Afterwards, a rather simple back and forth puzzle involving a specific configuration of these constraints allowed us to come to the conclusion that this could be settled over the matter of a quite fair deal for the lowest possible price. There was a lingering thought in our minds that possibly asking the development side for the absolute minimum they would settle for, and then allocate that extra percent to the recreation party to drive down costs, though, it seemed immoral. We are certain we have found the best and most efficient way to allocate the land fairly and respectively, though that proposal is configured and formulated upon math. We wonder if there is a more efficient way that disinvolves numbers and constraints that could drive the price down lower and still satisfy the people. Both sides want as much allocated land as possible, but obviously to drive down cost for extra benefit would require us to downplay the development side as they are the more cost-demanding supporters, we believe that in lieu of obtaining information through a survey or asking of sort from the people, we must hold true to the plan we have formulated with numbers. So with that out of the way, here’s the plan. 225 of the 300 acres of ranchland will be allotted to the recreational party, and the remaining 75 will be allotted to the development party. Secondly, 75 of the 100 acres of the old army base will be assigned to the development party, and the residual 25 acres will be accredited to the recreational party. Finally, the remaining location- the Boston mine will be entirely allocated to the development party; 150 acres. Therefore, in total, the development party will receive 75 acres of ranchland, 75 acres of the army base land, and all 150 acres of the old Boston mine. In turn, the recreational party will receive 225 acres of ranchland, and 25 acres of the army base land. Now obviously we could simply ignore this and drop to 50-50, because as far down as the development party is willing to drop, the lower the cost will be. Though at this point, efficiency is a major factor as well. Shall we move forward with this proposal?
Write Up
When in the course of the endeavor to solve the Meadows or Malls problem, we came across several key fragments that built up the understanding we needed to solve this mind-boggling problem.
The corner-point principle is an essential part of this ultimate equation, and in this problem, it involves several quantities that must be determined- and are represented as variables. Those variables can then be represented using linear inequalities, allowing us to find the optimal solution that would result in the lowest cost. The problem also involved a large series of matrices- number tables that can be mathematically combined. Matrices came in handy to formulate the final constraints, as once we organized all the preset statistics, we could create constraints that we would later use to solve the problem.
The functionality of multiplying and adding matrices together can be confusing, as you have to multiply/add the first row by the columns of the second matrix, so we did end up using a calculator. One of the most important elements of the problem involving matrices was the inverse matrix, which allowed us to acquire answers from multi-step equations.
In the end, we used our 12 constraints (that we later diminished down to 9) and ordered them in certain ways to gain all the possible outcomes. Distribution of “land” for the best cost came down to a few, as many of the constraint configurations involved too large of totals- therefore being outside of the feasible region. At last, we came up with a reasonable cost, and a distribution proposal that would most likely satisfy both parties.
Reflection
Throughout the entirety of the year... I only ever gained any collaborative experience while at school. Within the period of full-online, it felt like nothing more than “here’s the facts, do this”, now I know that was never the intention, but without the physical interaction, it’s quite bland. And since we were full online for so long, it was quite impractical to work in groups, other than those assignments where we simply fill out a group document. I feel that my skills to collaborate as a whole have been kept at a pause throughout the span of online school, but being back I feel somewhat of a want to engage more immediately in groups when we have a group assignment. I believe I have gained some confidence and serenity when I participate in group discussions. I am very thankful for this and I hope it persists to make me more and more comfortable.
Full online was an attack upon my mental stability, most of it was just too easy for me to learn anything, so much was based on participation, and the few assignments that were very difficult, I could not pick up on. Thankfully, I was able to manage to pick up on at least the gradually increasing ultimate goal to solve the unit problem; I could add the few things that I did learn together to formulate an idea of what this was all about. After Spring break, when we finally came back, as many do I felt weary of the sudden change. Though, after a few days, I began to exponentially appreciate the time we were given to learn face to face, and I managed to send my confidence through the roof. And even though I did not understand a couple of key concepts, I managed to somehow collect the information I needed... as if by nothing. I’m not sure what happened, but when we came back I felt better than ever to succeed, and I am so glad that I have. I learned to strengthen my focus and use it to finish strong.
Unit 1- The Orchard Hideout
Unit 1 Portfolio
Cover Letter
Geometry defines the 2nd and 3rd dimensions, as well as the relationship between the 2 dimensions. This unit focuses on the relationship between many of the small attributes of geometric figures and equations.
There were a lot of circles involved in this unit, but even then, we had to utilize the Pythagorean Theorem to solve the problem. The Pythagorean Theorem allowed us to look into the coordinate plane in which the overall circle rested on, we needed to determine exact distances from point to point and to calculate midpoints and distances between the smaller experimental circles that we created within the larger circle. Coordinates were the most important factor in the problem because they gave us a basis of symmetry, coordination, proportion, and of course location. The overall idea is to use the simpler methods and equations to solving circle attributes to find the overall solution.
Something else we had to understand was the square-cube law, “When an object undergoes a proportional increase in size, its new surface area is proportional to the square of the multiplier and its new volume is proportional to the cube of the multiplier”. If a cube is multiplied by m the surface area will be the original surface area x m2, and the volume will be the original volume x m3. The square-cube law was utilized to understand the expansion rate of the variables; the variables were also cylindrical, which made it even more complex. We measured a lot of circumferences and a lot of areas, so we needed to know what kind of expansions in the radius were happening across the period of time.
The final factor to the bewildering problem is proof, and not just true or false proof. Evidence. That’s right, we need evidence to show that this equals that. Throughout a series of graph analysis, we came to the conclusion that data can be used in situations where there are lots of variables and the conclusion is based upon an overall prime location on the graph. A scatterplot graph will have 2 axes with 2 topics but almost never the same points, and by utilizing graphs like such, we can find “lines of best fit”, and we can determine the “acceleration” or “deceleration” of the graph’s data if any. These 2 resources relate to the unit problem because there is a rate of change that must be determined, and there is a line of best fit to determine the last line of sight- the best viewpoint between the variables.
Unit Problem Description
The Unit Problem is based on the geometry of circles, and this problem took a lot of circum-navigation to get through. It starts with Madie and Clyde, 2 people who live on a circular piece of land with a radius of 500 feet. They decided to plant an orchard, and this orchard is based on a coordinate plane with a circle’s center at the origin of the coordinate plane origin. As the trees grow, Madie and Clyde wonder how long it will take until the trees grow wide enough to where they can’t see out of the orchard in any direction if they are standing at the exact center (0,0).
To solve this minimum of time, we need to know at what coordinates each of the trees were planted, what the radius growth rate of the trees are per month/year/decade, and the last possible line of sight from the center. After a while we were given our essential information and were about to go about solving the problem. At the time when Madie and Clyde wondered about how long it would take, the tree trunks each had a circumference of 2.5 inches. The cross-sectional area of the tree trunk increases by 1.5 square inches every year. The unit distance is 10 feet (the trees are planted each 10 feet apart). And the last line of sight is the line that goes from the origin through the point (25, ½). With these key resources, we can solve it. My process included a lot of translations and reconfigurations, but here it is:
Current size... C: 2.5 in. D: 0.795775 in. A: 0.49736 in2
Increase/yr... A: 1.5 in2 D: 1.38198 in. C: 4.3416 in.
50 units x 10 ft. = 500 ft. = 6,000 in.
4.99ft. is the last radius of a circle that therefore presents the last line of sight;
4.99ft. = 59.99in.
Using the data stats above, if the C-S area of the tree trunk increases by exactly 1.5 in2/yr, that would mean that the diameter increases by ~1.38198 in/yr. By dividing 59.99 by 1.38198, we can theoretically conclude that it will take approximately 43.4159683932 years until the orchard becomes a true hideout, maybe another 0.00025 or something years until the tree bark actually begins to fuse.
So apparently I was off, by quite a large amount, and I’m still not sure where I went wrong or how else to do it, but the correct answer was apparently 11.73, I and I can’t comprehend the process of getting there.
So in the end, the correct solution states that it will take 11.73 years until Madie and Clyde could not see out of the orchard anymore standing in the exact center. So they will have to wait, but not for too long.
Selection of work
https://docs.google.com/document/d/1MkjLUpNqkUWp-txudyxJd-SUx_92bCpp-0u_mfHglb0/edit
https://docs.google.com/document/d/123U6cLgDph9UAmhQUTu9W4iMEeJJC5oQ8W7il7L6jmM/edit
https://docs.google.com/document/d/1nG5xMYbQ3QVxoufEKIsuWmlaHD0aL1DKiurG64TTMeA/edit
https://docs.google.com/document/d/1OF6F3jZrn-fL2UkFHj8I5pJT7k8bz5CdiLUcvfeqqJg/edit
https://docs.google.com/document/d/1P-ntzGulzXqRgva7KhPToNh6Xo33GSFxMwRzIc7xy_0/edit
https://www.nytimes.com/2020/10/15/learning/whats-going-on-in-this-graph-climate-threats.html
Reflection
The idea of this problem was to better our understanding of the relationship between algebra and geometry, and I think it worked. I have previously used algebra and geometry in the same problem last year in advanced physics class, and this gave me a chance to revisit and advance with the concept. As a computer enthusiast, it’s obvious that I must use more than one system to complete a professional project, and I always have to think about the outcomes that could put me back at square one.
I also enjoyed this project because for once, a math equation had a correct answer, but there was a loophole. I technically didn’t end up with the correct solution, but knowing that the tree type was not specified kept me confident and less bewildered. I would like to do a project like this again because I feel like I really got to use ALL the resources that I’ve learned across the past 11 grades. This project, though frustrating at times, was quite insightful and really got me thinking about how precise something can really be. I definitely feel like I better understand the relationship between algebra and geometry now since I have come to use them back and forth all throughout the unit. The 2, though different, can make other outcomes and solutions so much more complex and precise that it really makes you feel like you just solved something so elaborate... but then it all makes sense... or not. It’s really cool, and I’d love to do another one of these projects where maybe we have to find multiple possibilities to a single problem.
Cover Letter
Geometry defines the 2nd and 3rd dimensions, as well as the relationship between the 2 dimensions. This unit focuses on the relationship between many of the small attributes of geometric figures and equations.
There were a lot of circles involved in this unit, but even then, we had to utilize the Pythagorean Theorem to solve the problem. The Pythagorean Theorem allowed us to look into the coordinate plane in which the overall circle rested on, we needed to determine exact distances from point to point and to calculate midpoints and distances between the smaller experimental circles that we created within the larger circle. Coordinates were the most important factor in the problem because they gave us a basis of symmetry, coordination, proportion, and of course location. The overall idea is to use the simpler methods and equations to solving circle attributes to find the overall solution.
Something else we had to understand was the square-cube law, “When an object undergoes a proportional increase in size, its new surface area is proportional to the square of the multiplier and its new volume is proportional to the cube of the multiplier”. If a cube is multiplied by m the surface area will be the original surface area x m2, and the volume will be the original volume x m3. The square-cube law was utilized to understand the expansion rate of the variables; the variables were also cylindrical, which made it even more complex. We measured a lot of circumferences and a lot of areas, so we needed to know what kind of expansions in the radius were happening across the period of time.
The final factor to the bewildering problem is proof, and not just true or false proof. Evidence. That’s right, we need evidence to show that this equals that. Throughout a series of graph analysis, we came to the conclusion that data can be used in situations where there are lots of variables and the conclusion is based upon an overall prime location on the graph. A scatterplot graph will have 2 axes with 2 topics but almost never the same points, and by utilizing graphs like such, we can find “lines of best fit”, and we can determine the “acceleration” or “deceleration” of the graph’s data if any. These 2 resources relate to the unit problem because there is a rate of change that must be determined, and there is a line of best fit to determine the last line of sight- the best viewpoint between the variables.
Unit Problem Description
The Unit Problem is based on the geometry of circles, and this problem took a lot of circum-navigation to get through. It starts with Madie and Clyde, 2 people who live on a circular piece of land with a radius of 500 feet. They decided to plant an orchard, and this orchard is based on a coordinate plane with a circle’s center at the origin of the coordinate plane origin. As the trees grow, Madie and Clyde wonder how long it will take until the trees grow wide enough to where they can’t see out of the orchard in any direction if they are standing at the exact center (0,0).
To solve this minimum of time, we need to know at what coordinates each of the trees were planted, what the radius growth rate of the trees are per month/year/decade, and the last possible line of sight from the center. After a while we were given our essential information and were about to go about solving the problem. At the time when Madie and Clyde wondered about how long it would take, the tree trunks each had a circumference of 2.5 inches. The cross-sectional area of the tree trunk increases by 1.5 square inches every year. The unit distance is 10 feet (the trees are planted each 10 feet apart). And the last line of sight is the line that goes from the origin through the point (25, ½). With these key resources, we can solve it. My process included a lot of translations and reconfigurations, but here it is:
Current size... C: 2.5 in. D: 0.795775 in. A: 0.49736 in2
Increase/yr... A: 1.5 in2 D: 1.38198 in. C: 4.3416 in.
50 units x 10 ft. = 500 ft. = 6,000 in.
4.99ft. is the last radius of a circle that therefore presents the last line of sight;
4.99ft. = 59.99in.
Using the data stats above, if the C-S area of the tree trunk increases by exactly 1.5 in2/yr, that would mean that the diameter increases by ~1.38198 in/yr. By dividing 59.99 by 1.38198, we can theoretically conclude that it will take approximately 43.4159683932 years until the orchard becomes a true hideout, maybe another 0.00025 or something years until the tree bark actually begins to fuse.
So apparently I was off, by quite a large amount, and I’m still not sure where I went wrong or how else to do it, but the correct answer was apparently 11.73, I and I can’t comprehend the process of getting there.
So in the end, the correct solution states that it will take 11.73 years until Madie and Clyde could not see out of the orchard anymore standing in the exact center. So they will have to wait, but not for too long.
Selection of work
https://docs.google.com/document/d/1MkjLUpNqkUWp-txudyxJd-SUx_92bCpp-0u_mfHglb0/edit
https://docs.google.com/document/d/123U6cLgDph9UAmhQUTu9W4iMEeJJC5oQ8W7il7L6jmM/edit
https://docs.google.com/document/d/1nG5xMYbQ3QVxoufEKIsuWmlaHD0aL1DKiurG64TTMeA/edit
https://docs.google.com/document/d/1OF6F3jZrn-fL2UkFHj8I5pJT7k8bz5CdiLUcvfeqqJg/edit
https://docs.google.com/document/d/1P-ntzGulzXqRgva7KhPToNh6Xo33GSFxMwRzIc7xy_0/edit
https://www.nytimes.com/2020/10/15/learning/whats-going-on-in-this-graph-climate-threats.html
Reflection
The idea of this problem was to better our understanding of the relationship between algebra and geometry, and I think it worked. I have previously used algebra and geometry in the same problem last year in advanced physics class, and this gave me a chance to revisit and advance with the concept. As a computer enthusiast, it’s obvious that I must use more than one system to complete a professional project, and I always have to think about the outcomes that could put me back at square one.
I also enjoyed this project because for once, a math equation had a correct answer, but there was a loophole. I technically didn’t end up with the correct solution, but knowing that the tree type was not specified kept me confident and less bewildered. I would like to do a project like this again because I feel like I really got to use ALL the resources that I’ve learned across the past 11 grades. This project, though frustrating at times, was quite insightful and really got me thinking about how precise something can really be. I definitely feel like I better understand the relationship between algebra and geometry now since I have come to use them back and forth all throughout the unit. The 2, though different, can make other outcomes and solutions so much more complex and precise that it really makes you feel like you just solved something so elaborate... but then it all makes sense... or not. It’s really cool, and I’d love to do another one of these projects where maybe we have to find multiple possibilities to a single problem.