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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.