The Philosophy of Geometry – Math Essay
I was recently assigned a Graduation Standard to complete. The Graduation Standard called for me to find an object in my community and figure out the objects height by using two different methods. The object I picked was a large pine tree. One method I used to find the height of the tree was the concept of similar triangles. In order to find the height with similar I needed to use my Hipsometer I made. A Hipsometer is a device that finds the angle of elevation.
The other method I used was Trigonometry. With Trigonometry I needed to measure the length of the trees shadow and the length of my shadow. To see diagrams of the methods locate and look over each diagram I have drawn of the situation.
The pine tree I used to work with was very large. So you can imagine how troublesome it was trying to measure the length of its shadow. Once I did have the length of its shadow I recorded it and had my mother measure the length of my shadow. My shadow was 15.7 ft while the trees shadow was 50.4 ft. I was supposed to find the length of the tree with these two measurements (and my height of 6 ft.) I found the length of the tree by first changing all of my measurements into inches, I did this by dividing all of the measurements by 12. Once I had the real lengths I used this formula…Height of Tree/My Height=Shadow of Tree/My shadow. That came out to look like this H/72=604/187. I cross-multiplied that equation and got 72 x 604=43488 and H x 187=187H. I then took 43488 and divided it by 187 to get 232.56. Then I needed to convert 232.56 back to feet so I divided it by 12. My final number was 19.38. The height of the tree came out to be 19.38 (again you can see the step by step process written on data sheet #5.)
After completing the height using Similar Triangles, I was to find the height of the tree-using Trigonometry. I needed the height from my eyes to the ground (5.9 ft,) distance from me to the object (21ft,) and the measure of the angle of elevation (27 degrees found with my Hipsometer.) I took those measurements and added them into this formula…tan X = H/Distance from Tree (X=measure of angle, H=Height of section of tree and my height. That formula looked like this…21tan27. It came out to 10.7. I added 5.9 (my height below eyes) to the 10.7 and got 16.4 ft. That was the height I found by using Trigonometry (check data sheet #5 for step by step process.)
When I had finished with both of the heights I became a bit angered because I noticed that the heights are not the same! Trying to figure out what I had done wrong, I soon realized that they should be different. My reasoning for this was that the shadow of the tree and I were both very inaccurate. Depending on the time of day and where the sun is located in the sky determines the length of the shadow. So obviously the shadows were either too large or too small. I think I should have measure the tree at 12:00 so I could get it at its peak position to obtain the most accurate shadow length. Also the width of the tree can effect the measurement of its shadow. The difficulty of using each method is that you will get a different measurement for the lengths of your object each time you do it. The Trigonometry would probably work best with something that is just straight up and down and does not move around or sway easily and something that is defiantly taller than you are. The Trigonometry can also be used anytime as appose to only be able to use it when the sun is out. The Similar Triangles would also work well with an object that is stationary but I think it would be better to use with an object that is limber or easily moved because all you need for Similar Triangles is the objects shadow. Such objects as towers, buildings, stadiums, and mountains should be used with Trigonometry. While trees, bushes, telephone polls, and antennas should be used with Similar Triangles.