Maya's Digital Portfolio

## COver letter

In this unit, we learned about the Pythagorean theorem and coordinate geometry. The Pythagorean theorem is the theory that goes with all right triangles. The main goal of this theorem is to find the hypotenuse of a right triangle. The formula for this is a²+b²=c². We used this formula a lot at the beginning of this problem. We used this a lot when finding the last line of sight in the orchard. Coordinate geometry is also a huge part of this. When I say this I’m talking mostly about the midpoint and the distance formula. These formulas were extremely important when solving the unit problem. The midpoint formula is (X1+X2)2, (Y1+Y2)2. This will help you find the midpoint between the two points. I used this when solving Proving Distance Part 2. I also used the distance formula. Which is, (X1-X2)²+(Y1-Y2)². Both of these formulas played a huge role in Proving the distance. At first, I didn’t understand these but when Julian drew it on a graph and showed why we do these equations and how we use them it clicked.

When talking about the square-cube law, there are many concepts that go along with it. For example, finding the circumference, finding the radius, finding the volume, and finding the surface area and the area. The equation for the circumference is C=2r. You can find the radius by calculating the midpoint of the circle and then using the distance formula to find the length of the radius. The volume equation is, V=h(r²). The equation for the area of a circle is A=2r. In this unit problem, we used all of these equations in order to find the radius of the trees and How to align all of the trees so that it would become a true orchard hideout. When understanding these equations and rules, the activity that helped me the most was when we put all of them together in order to find the size of the trees and how it becomes a true hideout and how long it takes.

Throughout the unit I learned a lot about geometric proofs. When solving a problem in this unit we always needed to make sure that the answer was true in order to make sure that it was accurate. When looking at different data in this unit we could conclude that some data may not be accurate based on certain variables. Some of those variables may include, lack of explanation, lack of research, or lack of information. We specifically looked at some graphs in order to prove whether they were accurate or not. I would say that about 50% of the time the graphs were not accurate. When learning about geometric proofs I first learned that some geometry and data may not be what it is supposed to. I then learned that there were certain equations that we could go through in order to prove things. As shown in the previous paragraphs. I finally learned how to look at data and figure out what was wrong and what I needed to prove. There were many activities that played an important role in finding geometric proofs. I would say that the major one for me was looking at graphs that we pulled from certain research and determining whether they were accurate or not. This made me understand that most math needs to be proved.

## Solution

When discussing the unit problem we need to think about the unit question. How long will it take for this to become a true orchard hideout? This means that all of the lines of sites would need to be blocked by the tree's radius, this is including the last line of sight. In order for this to happen all of the trees would have to grow enough for the trunks to be big enough to not touch but to block any lines of sight. There are 50 trees in the radius orchard and all of their radiuses are the same. This would mean that it would take a certain amount of time for each tree to reach the radius to make it a true orchard hideout.

The information that is important to know regarding this problem is, there are 50 trees in the orchard and each tree is planted 10ft away from each other, or 1 unit away from each other. The trees grow at a rate of 1.5 in² per year. The area of the trees when they were first planted is 0.5in². When starting this problem I needed to look at the last lines of the site and figure out which one would be the most accurate. In order to do this, I looked at a graph of about a quarter of the orchard. I then drew which line could be a good last line of site which is the black line in diagram one. You will see which line of site was best in diagram one. In the diagram, you can also see all of the last lines of the site.

The way that I figured out the rest of this problem was I looked at the starting area of the trees which was 0.5in² and then figured out the ending area of the trees based on the growth of the trees each year. I then figured out that the finishing area of the tree was 18.09in² You will see this in diagram two. When I had found both the finishing area and the starting area I subtracted the finishing area from the starting area and then divided them by 1.5 which is the rate of growth per year. This made me find how many years it would take the orchard to become a true hideout.

The solution to this problem is that it would take around 11.7 years for the orchard to become a true hideout. This problem had a lot of variables that went along with it. However, Maddy and Clyde will have a perfect hideout around 11.7 years.

## Reflection

Overall this unit was hard for me, I think that I could have done a lot better on this unit.The big thing that was hard for me about this problem was that we were online so I had to do a lot of it by myself. I had a very hard time when an assignment was given to me without a lot of knowledge on how to do it. I will say that over the zooms Julian did give us a lot of help on how to do the assignments and what formulas we were using to complete them. I personally have a very hard time paying attention to the zoom calls just because I am by myself and it's hard to just stare at a screen listening to someone virtually. I do understand that these are the circumstances that we do have to line in right now due to COVID. I think that I would have had a very different understanding of this problem if we had been in regular school. I think that in the future I would like to try and pay a lot more attention to the zoom calls and try to get some extra help from Julian individually. I will benefit from this because I think that I will actually learn things and be able to do and teach them on my own. This problem was done under very hard circumstances like I said I could have done a lot better, however, I am still proud of myself for getting the work done.

Although I didn’t do as well as I could have done on this assignment I think that there were still some things that I did really well when completing this assignment. For example, when I was feeling stuck on something I would reach out to my peers and ask them if they understood it and could explain it to me in a way that I understood. I also tried to reach out to Julian as much as I could. I will say that I haven’t asked for his help as much as I should have since we have gone full online. Something that I also was able to do really well in this unit was if I was stuck on something I looked back at some of the other assignments that I understood and tried to see if I could make any connections between the previous assignments to the current assignment that I was working on. This helped me a lot because I could almost try and teach myself how to solve the problem. When completing this assignment there were things that I did do good and I think that I didn't give myself enough credit.

Overall, this unit was hard for me. There were things that I did good on and there were things that I could have done better on. But I am proud of the work that I have created and I am very proud that I got all of it done.

## Write up

Throughout this unit, we have learned a lot of different math. From linear programing to 3D graphing, it has been a wild ride. Meadows or Malls is all about the cities dilemma. Durango is struggling with a land issue. The city is trying to figure out how much land goes to what use. There are six variables in this equation, each one representing a piece of land and its use. Rr is the number of acres of ranch land used for recreation. Ar is the number of acres of army land to be used for recreation. Mr is the number of acres of mining land to be used for recreation. Rd is the number of acres of rance land to be used for development. Ad is the number of acres of army land to be used for development. Md is the number of acres of minging land to be used for development. The end goal of this is to find the cheapest allocation of land that satisfies the constraints. To start off at the beginning, we started with the system of linear equations One of the pieces of work in this unit would be “Solving Systems of Linear equations: From Two variables to Three.” This was a very important part of the unit problem because it helps us understand how to solve a linear equation with a lot of variables. The unit problem has a lot of variables. By solving linear equations helps us find the constraints of the unit problem. Witch are:

1. Rr+Rd=300

3.Mr+Md=150

5.Ar+Mr200

6.Ar+Rd=100

7.Ar≥0

8.Mr≥0

9.Md≥0,

All of these constraints are the main thing that we needed to solve the unit equation. The constraints then form a feasible region will help us find the corner points and define the feasible region. The profit equation is then used to find the best answer by using the corner points.

This is where all of the numbers in the profit equation come from.

The second aspect of the unit problem is graphing, specifically finding the corner points. We didn’t particularly use graphing when figuring out the unit problem because there is no specific way to graph in six dimensions. Knowing this we had to figure out how to solve this problem without using a graph but still get to the corner points. To do this we used matrices. Matrices make it a lot easier to use a lot of variables and still find all of the corner points. This could be in place of using linear equations or graphing. However, when attempting to start this unit we learned how to do 2D and 3D graphing, this helped us use the constraints to find the final answers to some of the assignments. Also in this unit, we looked at planes, planes have a lot to do with graphing because they show us how the points lay on the graph. This can be done with 2D and 3D graphs. However, because there were six variables in this problem we needed to try a different method, matrices. We had to put together nine matrices using different combinations of constraints we created using linear programming. Six of those matrices did not meet all of the constraints. There we three matrices left that worked. After multiplying all of the matrices the best one that worked was the third combination of constraints or the third matrix.

Possible Combinations:

1. 1,2,3,4,5,6
2. 1,2,3,4,6,7
3. 1,2,3,4,6,8
4. 1,2,3,4,6,9
5. 1,2,3,6,5,7
6. 1,2,3,6,5,8
7. 1,2,3,6,5,9
8. 1,2,3,6,7,8
9. 1,2,3,6,9,9

This combination of constraints created the best matrix that had the best answer. This was the best price for the given land, costing \$353,750.​

## reflection

My junior year has been the hardest year I have ever had school-wise. COVID of course did not make this easy. However, I have had to learn new ways to make sure that I stayed on track with school and kept good grades. I have never had a school year where I couldn’t get all of the help that I needed from the teachers. Online learning is really hard for me because I need a lot more clarification than a lot of other students. When doing online learning I tend to drift off and get distracted easily because I am at home. I have had to learn how to use resources to help me with my assignments. COVID has been really tough on me socially as well. Especially because I am friends with most of the seniors at Animas. But because I haven’t gotten to see much of them. I have had to re-connect with all of the kids in my pod that I used to be friends with. They have been really helpful school wise and they are awesome friends. Working in groups in my pod can be chaotic at times but we always get what we need to do. I am in the crazy and the loud pod but honestly with me being crazy and loud that is where I belong. The thing that has saved me this year has been all of the support of the teachers. All of you guys have been very understanding and helpful when it comes to school and life in general. I have also gone through a lot mentally this year that has really torn me down. But all of the teachers at Animas have been so helpful and understanding. I am happy with how I have done this year under the circumstances. I never would have thought that I could work this hard in school and put in this much work. All I can say that thanks to the support of my friends, family, and teachers, I am proud of myself.