Meadows & Malls Unit
Report:
In this problem we were assigned the role of a consulting firm. We were given the task of finding the best possible solution of dividing 550 acres of land, for either recreation or development.
Our variables are as follows:
In this problem we were assigned the role of a consulting firm. We were given the task of finding the best possible solution of dividing 550 acres of land, for either recreation or development.
Our variables are as follows:
- The variables shown below represent the amount of land that is designated towards either recreation or development.
- RR is the number of acres of ranch land to be used for recreation
- RD is the number of acres of ranch land to be used for development
- AR is the number of acres of army land to be used for recreation
- AD is the number of acres of army land to be used for development
- MR is the number of acres of mining land to be used for recreation
- MD is the number of acres of mining land to be used for development
Parcel
Improvement costs per acre for recreation
Dalton Ranch: $50
The Old Fort:$200
The Boston Mine:$100
Improvement costs per acre for development
Dalton Ranch:$500
The Old Fort:$2,000
The Boston Mine: $1,000
Constraints:
In order for us to come to a solution for this problem, all the constraints must be satisfied.
Improvement costs per acre for recreation
Dalton Ranch: $50
The Old Fort:$200
The Boston Mine:$100
Improvement costs per acre for development
Dalton Ranch:$500
The Old Fort:$2,000
The Boston Mine: $1,000
Constraints:
- RR + RD = 300
- AR + AD = 100
- MR + MD = 150
- RD + AD + MD ≥ 300
- AR + MR ≤ 200
- AR + RD = 100
- RR ≥ 0
- AR ≥ 0
- MR ≥ 0
- RD ≥ 0
- AD ≥ 0
In order for us to come to a solution for this problem, all the constraints must be satisfied.
The table above shows all the possible combinations of constraints we can create. After solving this by combining our constraints with the cost per acre, we can find a solution that is within the feasible range. For this particular problem the best possible solution would be 1,2,3,6,4,9. This solution is the best at minimizing the amount it costs to improve each piece of land. This solution also satisfies all the constraints, as well as fits within the feasible region. If we were to go ahead with another solution we would increase our chances of not only spending more money, but also not having well distributed land and running the risk of having unsatisfied clients. Another solution that is close but not as reliable would be 1,2,3,6,4,8. This solution is within the feasible region, and doesnt cost much more than the previous solution. Although if we were to choose this solution the biggest change would be that there is 50 acres of land designated for mining land and 0 army land designated for recreation. This means that the land used for recreation is more likely to fall on army land.
Write up:
“The Corner Point Principle” & Feasible regions
When we are trying to find/solve for the best possible solution in a two-variable equation and or three variable equation we will always be working with constraints. There will also be a feasible region. The feasible region will help identify the points that satisfy all constraints. The two different types of equations will create two different models or shapes. A two variable equation will always create a 2-dimensional model, and a three variable equation will create a 3-dimensional model. As shown in the document attached below, Corner points will always be the most reliable and or best solution at maximizing the sum.
https://docs.google.com/document/d/1Tjdp3TY88IvgIYRQS-i5ilnQoIgDpUu5vbrreLZ2ZoE/edit
Three Variables
A three variable equation is representational of the three dimensions within an equation. We know we can solve for intersections in a two variable equation, and just as similarly we can solve for an intersection in a three variable equation. This process is very similar to that of solving for an intersection in a two variable equation. The only difference is a couple extra steps you have to complete, as well as three possible solutions which is discussed below. To solve a three variable equation all you have to do is compare two of the three equations, eliminate one variable from each, and then use those two equations to solve for the variables. As an examples lets say we were given the following system of three variables,
The graph would like the following for this particle equation:
Three Solutions
As mentioned previously a system of three variables has three possible solutions, also known as intersection points. In some cases there is only one solution, as well as the possibility of infinite solutions. If there is no solution, this means that the lines are parallel and never intersect with one another. In this case there will be no variables to apply and therefore solve. Shown below are representations of the three possible solutions both in a 3-D model as well as written form.
As mentioned previously a system of three variables has three possible solutions, also known as intersection points. In some cases there is only one solution, as well as the possibility of infinite solutions. If there is no solution, this means that the lines are parallel and never intersect with one another. In this case there will be no variables to apply and therefore solve. Shown below are representations of the three possible solutions both in a 3-D model as well as written form.
Matrices:
Using matrices in math can really help organize and keep your work clean. A matrix is a way to split up systems of equations into values and variables. The process of solving a matrix is shown below, you can do this both with and without a calculator, both examples are shown in the images below. We start to solve by separating the values and variables. In matrix number one we would have the values of 1,2,3, and 4. In matrix number two we have the x and y values, or in our equation the values of w and z. In the last matrix we have the solutions of the equations, which in our case would be 6 and 16. Now we can solve this two ways, one by elimination and working through the equation by hand, or we can plug it into the calculator which will find the inverse, as I did shown below.
Using matrices in math can really help organize and keep your work clean. A matrix is a way to split up systems of equations into values and variables. The process of solving a matrix is shown below, you can do this both with and without a calculator, both examples are shown in the images below. We start to solve by separating the values and variables. In matrix number one we would have the values of 1,2,3, and 4. In matrix number two we have the x and y values, or in our equation the values of w and z. In the last matrix we have the solutions of the equations, which in our case would be 6 and 16. Now we can solve this two ways, one by elimination and working through the equation by hand, or we can plug it into the calculator which will find the inverse, as I did shown below.
Reflection:
- What growth have you noticed so far during your junior year in your collaborative skills, and how did these changes affect your ability to work with your group members?
- How did your experience with remote and hybrid learning during this unit help you to grow as a student and mathematician, and how did it affect your ability to be successful in Precalculus and in completing this project.
Portfolio
-Cover Letter-
Pythagorean Theorem & Coordinate Geometry:
The Orchard Hideout Unit covered things such as distance. One of the main focuses of this unit was learning the equation a^2 + b^2 = c^2. This equation is called the Pythagorean Theorem. In this equation, C would equal the hypotenuse. A and B would both represent the sides of a right triangle. The basic understanding of this equation is that the area of a square’s sides creates the hypotenuse, which is equal to the area of the squares on the opposite side. This equation was one of the main focuses of our unit. Applying this was critical to solving work such as POW 2 Fire! Fire! Within this work, we had to identify the midpoint and distance of the fire station from the road(s). To identify the distance we must do the equation (x1 - x2)^2 + (y1 - y2)^2. We then had to find the midpoint. To do this we had to find the average of both x coordinates and y coordinates, and then divide them by two. In doing all of this we were able to identify the best possible location for the fire station, which was an equal distance from one, two, three, and four roads. Another example of these ideas being shown is in our classwork “Down The Garden Path.” This work played with the same idea as Fire! Fire! In this problem, Leslie wanted a path that was equidistant from two flowers in her garden.
Circle & The Square Cube Law:
The circle and Square Cube Law is the relationship between length, surface area, and volume that determines the possible size or change in the size of an object. Throughout this unit, we continued to apply this concept. This knowledge allows us to find the difference between volume and area. It is first important to differentiate the difference between area and volume. The area often refers to a 2D or flat object. Volume is a 3 Dimensional object. The “Cylindrical Soda'' classwork is an excellent example of these ideas being applied.
Proof:
Throughout this unit we discussed how crucial proof is within a well throughout math solution. We learned that proof is the largest building block to creating reliable solutions and work. This concept was applied to almost every piece of work we did. Julian drilled it into our heads just how important it is to be able to back up your math work with proof. The two main examples that come to mind for this are POWS. The whole point in both POW: Fire! Fire!; and the POW we did earlier in the year was to dive deep into exercising this idea of proving our solutions. Throughout this unit, I learned that proof is not only a fundamental ingredient in creating and showing a truly beautiful and well-done solution, but it also gives you a lot of credibility within your work. From now on I can tell that proof is going to be a concept I weave into every project I do.
-Solution -
Introduction:
In the Orchard Hideout Unit, we were introduced to Maddie and Clyde. Maddie and Clylde wanted to create a Orchard with a hideout in the middle of the trees. The goal of this hideout was for the center to be empty and surrounded by trees so you could not see the outside world from the middle of the orchard. In order for this to be a true hideout, every possible line of sight of the orchard must be covered or blocked by a tree. Maddie and Clyde want the middle of the orchard to be a sort of sanctuary. To help achieve Maddie and Clydes goal, we were given the task of figuring out the length of time it would take for the trees to grow and cover these lines of sights, and how big the trees must become. In this problem we were given the prior knowledge of the orchard having a 50 unit radius. Now, we must find the area of the hideout, as well as the last line of sight.
Process & Justification:
To begin solving this problem I reflected on all the prior knowledge we gained throughout the unit. In this unit we learned, the radius of the orchard is 50 units, each unit being equal to 10ft, each tree's circumference is 2.5 and grows 1.5 inches every year. We also learned that the last line of sight will pass through the points of (25,½) and go through (50,1). Using the diagram below we can follow our next steps.
To answer the unit problem we must find out how long it takes and where the trees will reach our line of sight. Only then will the orchard become a true hideout. On either side of the line of sight there are two distinct triangles. Let's call these triangles A and B, as labeled above. Using our knowledge we can conquer that these triangles both share a 90° angle, meaning they are similar. This knowledge helps us, because now we know the units of one triangle, meaning we can find the units of the other as well; the triangles are similar! We know that triangle C is similar to that of triangle D. If we find the hypotenuse of triangle C we can find the units of triangle D. We can do this by identifying the hypotenuse of the triangle labeled “c.” In order to find the hypotenuse we must apply the Pythagorean Theorem. The equation to do so is a2+ b2=c2. This math being applied to our problem is shown below.
This would make our missing hypotenuse equal to the square root of 2501, or approximately 50.0099990002. We can conclude that the hypotenuse of triangle c is tangential to the circle. Using this knowledge we can now find the radius of the short leg. This is shown in the work below.
After we apply this math we now know that the radius is equal to 0.019996 units. We know that one unit is equal to 10 feet. Let's find the final area of the area of the tree. To do this we must know the starting area and the ending area. We must know this so we can find how many years it will take for the starting area to get to the ending area. Let's work on getting the starting area first! We know that the starting tree circumference is equal to 2.5 inches (C=2.5). Now we must apply the equation for circumference which is c=2r. So, 2.5=2r which gives us a total of r=2.52 or approximately 0.398 inches. After applying this to the area formula, A=r2, we can find an area of 0.4792inches, making this our starting area. To find our ending area, we know that because each unit is equal to 10 feet our radius is going to be equal to 2.4 inches (work shown below)
After plugging 2.4 into our formula for area (A=r2)We get an ending area of exaclty 18.0883 inches. The last step to find our final solution is subtracting our starting area (0.479^2) from the ending area (18.0883) to see how much the trees need to grow. This gives 17.591inches^2. Now we must divide how much it has to grow by the rate at which it grows, meaning we must do 17.591 inches^2 by 1.5 inches^2. This number will give us a total of 11.73 years.
Solution:
After Maddie and Cylde plant all of their trees in the orchard with a radius of 50. Maddie a Cylde will get their true orchard hideout in approximately 11.73 years, or 11 years and 8 months!!
-Reflection-
This fall semester we studied the Orchard Hideout Unit. While we learned half of this unit in person, we got pushed to learn the rest of this unit from the safety of our homes during these troubling times. I'm going to be 100% honest. I am very disappointed, frustrated, and overall very embarrassed with the work, or lack thereof, that I produced for this unit. I struggled significantly during this unit. This is very frustrating for me because I know I have the potential to be a better student. I am not a student who gets this far behind. I always stay up to date on my work and usually produce work that I am extremely proud of. During this unit, I simply did not understand the material. Lack of understanding made it difficult for me to be motivated. I have always struggled with math, even though I quite enjoy math when I can do it. Having to practically teach myself the material for the unit was very difficult. While I understand I could have reached out to Julian for help, and I am slapping myself for not reaching out more, I don't know if this would have helped me. I am a very hands-on learner. I like to be able to see things and work things out and bounce ideas off of my peers, but because of the situation of having to work from home, this was not an option. Another obstacle for me this semester was just honestly lack of motivation. At a certain point, I think I got so overwhelmed by the amount of work we were being asked to get done, as well as struggling with poor time management skills. Having to juggle all of my classes and make sure I had time to get everything done was extremely difficult for me. I am a very busy person. I have dance Monday, Tuesday, Thursday from around 5:30-8-8:30 at night. On top of this “The Polar Express” train started up and I don't get done with work from there until around 10 pm and I work any night I am not dancing. I also am getting ready for a huge trip to Flordia for a competition called “International Model and Talent Association” or IMTA. I have been preparing to go to this competition with my agency up in Denver for the last 18 months. This is an opportunity for me to get a head start in the career I dream of. I am not trying to make up excuses, but I honestly just got very overwhelmed with everything going on in my life. I think part of the reason I struggled so much this semester was just honestly me shutting down. In the middle of this unit, it felt like most of the work we were doing was “busy work” for lack of a better term. I understand that this was a method of trying to keep us engaged in the work we were doing from home. I am extremely grateful for the push that Julian gave me to get more motivated to get my work done. In no means necessary did he have to give me an extension, but I am grateful he gave me the opportunity to show the work I am truly capable of achieving.