Cover Letter Introduction Over the course of this unit, we explored many concepts to help us solve the unit problem, “How soon after they planted the orchard would the center of the lot become a true orchard hideout?” These concepts include the Pythagorean Theorem & Coordinate Geometry and Circles and the Square-Cube Law. Towards the end and throughout the unit, we came up with a lot of proof to solve the unit question.
The Pythagorean Theorem and Coordinate Geometry The Pythagorean Theorem is used to find the distance between two points on a plane, as well as finding perimeters, surface areas, and volumes of geometric shapes. This equation proves that if the lengths of any two sides of a right-triangle are known, the length of the third side can be found. The equation is a2 + b2 = c2. Another proof that we found for this was that when a triangle has an angle that measures 90 degrees, and squares are made on each of the three sides, then the largest square on the hypotenuse has the exact same area as both of the smaller squares put together. This shows that “a” and “b” from the equation represent the two smaller sides of the triangle (the legs), and “c” represents the longest side (the hypotenuse). Coordinate Geometry, also known as “Analytic Geometry”, is the study of geometry that uses coordinates to aid in geometric analysis. I found that we used this concept quite a bit while I was solving the unit problem. We used three points on a plain (three of the trees in the orchard) to create a right triangle, and then we had to find the distance between the three points in order to find the hypotenuse of the triangle so that we could find the last line of sight in the orchard.
Circles and the Square Cube Law The Square Cube Law is used in a variety of scientific fields, and it expresses the relationship between the surface area and volume as the size of a shape increases or decreases. As dimensions of a 3 dimensional shape increase, the volume will continue to grow faster than the surface area. In the assignment, The Square-Cube Law and Unpacking the Article, we were working with two different spheres. One of them had a radius of 2 cm, and the other had a radius of 10 cm. The formula for surface area is 4r2, and the formula for finding the volume of a sphere is 43r3 . When we plugged in the numbers, we ended up getting 50.24 for the 2 cm sphere and 1,256 for the 10 cm sphere. We then needed to find the volume of each sphere. The result for the 2 cm sphere was 33.41 cm, and the result for the 10 cm sphere was 4,186.67 cm. We also obtained the formulas for a circle: D= 2r, C= 2r, and A=r2. This assignment helped us better understand the relationship between surface area and volume in a 3 dimensional shape, and how strength scales with size. We learned that the surface area scales to the square of the radius, and that volume scales to the cube of the radius. Therefore, as the dimensions of a 3 dimensional shape increase, the volume will always continue to grow faster than the surface area.
Proof It's typically difficult to understand why something needs to be done or why anything means what it means unless you have some form of proof. Proofs aren’t only there to confirm something’s validity, they are important in confirming and developing our understanding of different concepts and explanations for why things are the way they are. Mathematical proofs are valuable to us because they offer new methods, concepts, and they make us exercise our logical reasoning and problem solving skills. When researching different topics and looking over statistics, it's common to come across articles that don't display the data in order to responsibly support an argument. It can be filled with bias or over exaggerated to make you believe something other than the truth. Sometimes the sources that were used to create the article can be extremely unreliable and biased as well. When we completed the assignment, Data Primer: Analyze Your Own Article, we learned how to find reliable sources on our own and analyze them to be sure that they were authentic. This helped us develop a better understanding of what responsible data presentation should look like. Out takeaways from the Data Primer mini-unit consisted of: when the data is being drawn from a legitimate source and it doesn’t integrate any outside source to make it seem biased, the graphs are put together in a way that makes sense and they don’t use anything that would try to draw you towards a certain conclusion, it lists all of its sources and doesn’t leave anything out, and it doesn’t use any biased authors and all of the data is supported, it is most likely a very trustworthy source. My understanding of a geometric proof is that it's a step-by-step explanation that uses definitions and previously proved theorems and equations to draw a conclusion about a geometric statement. I think that we used this type of proof when we watched the video, “But why is a sphere's surface area four times its shadow?”. In the video he explains exactly why the surface area of a sphere is four times its shadow. He proves his explanation in several ways as well. The only activity we completed that played an important role in my understanding of these ideas was the assignment, Data Primer: Analyze Your Own Article, because it was the only thing that felt relevant to me, so I think I just had less of a difficult time understanding it.
Unit Problem
Introduction This is the unit problem: “How soon after they planted the orchard would the center of the lot become a true orchard hideout? “. We’ve finally solved it after an entire semester of learning new concepts and formulas, and practicing them over and over again! We were given the takst to be able to solve this problem by the end of the semester using the Pythagorean Theorem and coordinate geometry, the Square-Cube law and circles, and different mathematical proofs.
Process and Justification One of the first important things we discussed having to do with the unit problem was line of sight. Keep in mind that the orchard has been planted on a circular lot with a radius off 50 units. We knew that our last line of sight needed to go through a specific midpoint of two trees, which we’re told is at (25, 1/2) so that it will eventually go through the point (50, 1). We’ve been using this reasoning throughout the unit in other assignments, we’re now just scaling it up into a bigger situation. The line of sight comes from the concept of the midpoint, and the final radius of our tree comes from the concept of the distance from a point to a line. This helps me understand that we’re looking for the perpendicular distance. Drawing out the lines of sight on the orchard creates two triangles, one with a right angle against the line of sight, and a larger triangle that follows the line of sight all the way out to the end of the orchard. The larger triangle is 50 units long on the base leg and 1 unit long on the vertical leg. The smaller triangle has a hypotenuse of length 1 and we need to find the length of “the hideout tree radius”, which is the length of the vertical leg. We know that the two triangles are similar because they both share the angle towards the center, and they both contain a 90 degree angle. So they are the same shape, just different sizes. We then used the pythagorean theorem to find the hypotenuse on the larger triangle, which was approximately 50.01 units. After all of this, we’re now ready to set up the ratio in order to find the length of “the hideout tree radius”, or r. The missing side of the smaller triangle (r) divided by 1 is the same thing as 1 (hypotenuse of the smaller triangle) divided by the hypotenuse of the larger triangle, which is 2,501 units. This means that r is 12,501 . Now we’ve determined that our hideout tree radius is 0.019996 units. We now need to find the areas of our starting tree and our ending tree. We want those two values because we know how fast our trees grow (1.5 in²/year). If we have our starting area and ending area, we can figure out how many years it took to get from one value to the other. We know that our starting tree has a circumference of 2.5 inches, so we can use that value to translate it into the radius of our starting tree, which is approximately 0.398 inches. We can then use this value to find the area of our starting tree, which is approximately 0.497 in². Now, if we look at our ending tree area, we know that the radius is 0.019996 units. we can take that value and multiply it by the fact that there are 10 feet in one unit, and then we can multiply that by the fact that there are 12 inches in 1 foot. When we completed this multiplication we were left with a radius of 2.4 in², which actually makes sense. With this information we can confirm that the ending tree area is 18.0883 in². We now have the starting cross-sectional area of the tree (0.497 in.²) and the ending cross-sectional area (18.0883 in.²) of the tree. we now need to find the difference between the two values, and we do that by subtracting one from the other, leaving us with 17.591 in². Now, all that’s left to do is divide how much it has to grow by how fast it has to grow. This gives us approximately 11.73 years, or 11 years, eight months, 22 days, four hours, 46 minutes, and 36 seconds.
Solution It takes approximately 11.73 years, or 11 years, eight months, 22 days, four hours, 46 minutes, and 36 seconds for the tree to grow 17.591 in², so that it will become a true orchard hideout.
Before this unit, I definitely wasn’t aware that there was really a strong relationship between algebra and geometry, but now I see that there is. Algebra uses variables in the forms of letters and symbols, and geometry studies points, lines and many other types of shapes and objects. While we were working through this unit, I realised that in many instances you actually need algebraic equations to learn things about certain shapes and solve complex problems. It’s all been extremely difficult for me to understand and comprehend, but in the end I think I eventually got a decent understanding of the unit problem using the skills I’ve developed. Though I did have to rewatch the video of Julian explaining how to come to a solution at least 23 times, I was able to complete a writeup for the unit problem. To be completely honest, I didn’t actually mind doing math at all while we were still at school in person, and that's very new for me. I was extremely motivated to learn in all of my classes, and I had straight A’s for the first half of this semester. I think that when we went online again it just felt too unexpected, and it really threw me off. I have a hard time learning when we’re not in class, and I don’t really understand things when I’m sitting behind a screen. I lost all of my motivation and I began looking at all of the negative things in your class, and the rest of my classes. Towards the end of the semester, I just completely gave up. Even over break, I would try to take out my laptop to work on the portfolio and I would immediately get so frustrated with it because I just had so much to do. I’ve been working on this assignment for about five hours now, so I’m extremely proud of what I’ve done. I definitely want to try to stay caught up next semester, and look at things in a more positive light. I’ve just been so frustrated because this pandemic has ruined so many things in my life, and I don’t always like to admit that for some reason. I’m really grateful that you haven’t given up on me yet. I have a lot of things to deal with all of the time and school has never been a priority. I don’t work well behind a screen for a variety of reasons and I’m not always great at reaching out for help or even finding the motivation to do so. I definitely want to work on that for next semester because I know I need to if I want to succeed. Something I’ve had to remind myself of during this unit is that I can’t control anything that happens around me, I can only control how I respond to it. I can’t control this pandemic, or the content you put out for us, or when we go back to school (or if we do go back at all). I can only try my best to adapt to it and work around my frustrations, so that’s what I’ll try to do next semester. Thank you Julian.