Kristina Bodewes 20 November 2020 Pre- Calculus p 2 Julian Springer Orchard Hideout Portfolio Cover Letter: Throughout the unit of “The Orchard Hideout” we learned many different mathematical concepts and ideas. One of the main topics which we covered include coordinate geometry by using the distance formula to find the distance between two points on a grid, the midpoint formula to find the midpoint between to points on a grid and the Pythagorean Theorem to find the length of any side of a triangle. Another concept we learned was the square cube law and the relationship between volume, area, surface area and perimeter. Finally, we learned how to “prove” our work in an organized and effective manner that clearly communicates our understanding and explanation for our final resulting products. The Pythagorean Theorem and Coordinate Geometry: This semester we furthered our understanding of the Pythagorean theorem: (a2+b2=c2) by investigating problems such as the one in our warm up from 11/03/20 where we were given two similar triangles with only certain dimensions given. We were to find the missing dimension of one of the triangles using the Pythagorean theorem. This formula proved to be important in the unit problem as well when we had to perform a similar process when finding the lengths of two triangle with the same angle measurements. In addition, we also reviewed the distance formula: d=(x2-x1)2+(y2-y1)2 in order to find the distance between two points on a grid. We used this in the very beginning of the unit when we investigated the sprinkler dilemma and where the best place to put a sprinkler system would be. In the same investigation we also reviewed the midpoint formula: mp=x1+x22, y1+y22 This formula was used specifically when trying to find the exact point at which a sprinkler which turned 360 degrees around would hit two flowers with the equal amount of water. Circles and square cube law: Over the course of this unit, we also learned how to use the circles and square- cube law in order to more deeply understand the relationship between how the volume, area and perimeter of an object, particularly circles, spheres and cylinders, scale as the “size” of the object changes are not straight- forward or linear. The square cube law causes the area of a cube to scale with the square of its length, and the volume to scale with the cube of its length. For example if we took a cube with the side lengths of 2 units, its area would be 4 because 22=4, and its volume would be 8 because 23=8. To analyse the situation further, we can take another cube with the lengths of 10 units and find the area; 100 units since102=100 and the volume which would be 1,000 since . We know that this is being squared and cubed because when we take the lengths; 2 and 10 units, they are translating by multiplying by 5 since 25=10. The area is being squared because 4 translates to 100 by being multiplied by 25 which is the square of 5. Finally, the volume is being cubed because 8 translates to 1,000 by multiplying by 125 which is the cube of 5. Proof: This unit we learned the effectiveness and importance of providing a proof for our work and our conclusions. Not only do proofs provide a pillow of additional reference and evidence for our findings, but they also help us to check our work and simply see the steps much easier; especially for more complex and rigorous geometric problems. The application of proofs was through finding the lengths of different sides of triangles, and perpendicular bisectors. We started the processes by finding what was “given”, and writing those findings in one collom, and “proving” that other things could be true after applying different concepts such as the pythagorean theorem to show that different sides were proportional to others, or had “similar” shape etc. In the unit problem itself, this was a very helpful way to organize my thoughts and evidence for the claims I was making based on my calculations. Unit problem: The unit problem stated that two people, Maddie and Clyde wanted to know how long it would take before they could no longer see out of their orchard from the very center given that it had a radius of 50 units, the circumference of each tree was exactly 2.5 inches to begin, the cross- sectional area of the tree trunks increased at a rate of 1.5 square inches per year, the unit distance [from (0,0 to (0,1)] is 10 feet, and the last line of sight goes from the origin through point (25, ½). The final objective therefore was to find the amount of time it would take for the orchard to become a “true orchard hideout” or when Maddie and Clyde could no longer see outside of the orchard when standing in the very center. Using the information given, we were able to begin the process of finding the specific amount of time necessary. Process and Justification: Using the information given by the situation, we can slowly use it to solve the problem. First, I set up the drawing of the orchard hideout using the point (25, ½) as the midpoint of the final line of sight to find that the actual line of sight ends at point (50,1) (note that I drastically shortened the scale on the x-axis to simplify the picture and process. I drew a line from the center of the orchard through both points (25, ½) and (50, 1) in order to create a slanted line. I then connected that line to the x- axis and created a right triangle. The big triangle that was created from this had a base leg of 50 units, and a vertical leg of 1 unit. I then created a smaller triangle using the same angle as the large triangle. This triangle had the hypotenuse of 1 unit. The next step was to find the length of the vertical leg of the small triangle (hideout tree radius) It was given that the two triangles were “similar” as they shared a right angle and the angle created from the line of sight and the x-axis. Therefor the third angle had to be the same as well. In order to find the missing length (hypotenuse) of the large triangle, I used the Pythagorean theorem; 502+12=2501; 50.0099990002. Knowing the hypotenuse, the next step was to rationalize the difference of r/1 to 1/2501. So the hideout tree radius would be derived from the equation r=12501 which is approximately 0.019996 units. The easiest way to go about applying the mathematical process in this case was to go through the list of information given and cross things off as I went along, and add the new information such as the newly found hideout tree radius to the list. The next step I took was creating a table that provided an organized way to communicate my next steps:
After finding the starting and ending area, I had to find the difference which was derived from the equation 18.088-.497=17.591in2which is how much the trees had to grow. The final step is to format the amount that the trees had to grow (17.591in2) and divide it by the rate at which they grow which is 1.5in2 per year. The final answer came out as 11.73; So, it would take the trees 11.73 years or approximately 11 years and 8 months to finally become a true orchard hideout.
Reflection: This problem used a mix of geometry and algebra. I now better understand the relationship between them from many part of the process of solving the problem. One that stood out to me the most was the use of the Pythagorean theorem to rationalize the similarity between two triangles and the steps I had to take in order to find the area of the trees to know how much they had to grow. I realize now how important it is to have algebra, after all we would not have been able to solve the problem without it, but also how important geometry is to actually visualize the concepts from algebra in a more in depth yet simpler way.
Throughout this entire unit however, I took away the necessity for the midpoint and distance formula. I still struggle to set up and fully understand the importance and meaning behind proofs, however this semester I was able to better conceptualize what exactly they do for the process of finding the relationship between two shapes.
I’m rather embarrassed by my work in this unit. This last quarter I have been so caught up in the unimportant things that I see behind my phone screen that I haven't been applying myself or trying as hard as I usually would or should in school in general. I think that this writeup is a perfect reflection of that as it’s 3:55, this is due in 5 minutes and I didn’t even look at the assignment until 4 hours ago. I think that realizing this now is likely a good thing because I know what I need to change in order to be more successful this coming semester.