Measure Your World Project
Overview
This project was a geometry based project. We discussed measurement, we began learning about two dimensional measurement, then went on to three dimensional measurement. To begin everything we quickly reviewed the Pythagorean Theorem (A2+B2=C2), then we moved on to the distance formula. Once we were done with that we began to look at the unit circle. To do this we used the radius and angel of the radius to create a right triangle, from that we were able to find the other side lengths of the triangle and ultimately were able to find the point where the radius intersected with the circle. After that we went on to define sine, cosine, and tangent. We also did a problem that talked about how early explorers found the location of Mount Everest. We were first told a story about how the explorers were mapping India, they saw Everest from a distance but could not get close enough to measure it. So instead they used two points and calculated the angle from those to the top of Everest. We were given the measurements and had to find the distance to Everest. I enjoyed that assignment because it took what we had been learning in class and applied it to a real world situation, it proved that what we were learning in class wasn't just busy work. Next we began something called the corral problem, it was about a rancher who wanted to build a corral, she could only afford 300 feet of fencing and wanted the biggest corral possible. We also had to find a formula to find the area or the rectangle with only one dimension, the formula ended up being X*150-X=A, X being the known side length, and A being the area. There were multiple parts of the corral problem, the first being with rectangles, the second being with triangles, the third with double the fencing, the one after that with a regular pentagon, and the final one being any polygon we wanted. We did this by separating the polygon into triangles based off the center point For the final part I decided to find the area of a 33 sided polygon, I wanted to challenge myself. We then began to look at volume, first we had to find the volume of a picture frame with 45 degree angles. The final assignment was finding the volume of a frustum (a pyramid with the top chopped off).
Design your own project
Next we had to design our own short project where we measured something. After much pondering I decided to measure the volume of Apple's new headquarters. I decided to measure this because I have a love for all things Apple, and its going to be a really cool looking building. I thought it would be a significant challenge since the top of the building will have an angle to it.
Since the building is round I originally decided to measure it as a cylinder, then Dr. Drew pointed out to me that there is no building in the middle. He suggested I unfold the ring and look at it as a long rectangular prism. To do that I used the circumference of the outer circle as the length, the width of the building as the width of the rectangular prism, and the height was the height of the rectangular prism of course. I also broke the angle of the roof out into two right triangles that I would be able to subtract from the total area of the building.
Since the building is round I originally decided to measure it as a cylinder, then Dr. Drew pointed out to me that there is no building in the middle. He suggested I unfold the ring and look at it as a long rectangular prism. To do that I used the circumference of the outer circle as the length, the width of the building as the width of the rectangular prism, and the height was the height of the rectangular prism of course. I also broke the angle of the roof out into two right triangles that I would be able to subtract from the total area of the building.
I began this project by looking online to find the dimensions of the building, through my research I was able to find the diameter, height and width of the building. I used the diameter to find the circumference. To do this I used the formula: Circumference = π * diameter. By doing this I found that the circumference of the building is 5,073.672135547516 feet.
Next I created a rectangular prism using the circumference of the building as the length, the height of the building as the height, and the width of the building as the width. I decided the ignore the curve of the roof for the time being I used the formula
Volume=Length * Height * Width, to find the volume. I plugged the dimensions into the formula and found a volume of 67,478,839.40278196 cubic feet. This may look like the final answer to the problem, but it is not. The roof of the building has a unique curve to it.
Volume=Length * Height * Width, to find the volume. I plugged the dimensions into the formula and found a volume of 67,478,839.40278196 cubic feet. This may look like the final answer to the problem, but it is not. The roof of the building has a unique curve to it.
Moving on to calculating the part of the roof that needs to be subtracted. When I was looking online I came across an image that showed the shape of the roof. By using this image I was able to break the slant of the roof into two rectangles (outlined in pink) then create two right triangles (outlined in blue) with the part of the rectangles the roof didn't occupy. When I drew out the right triangles I realized that I only had one side length to work with. I began to think about ways to look at the triangles in a different way. I began to think about the right angles on the triangles. Soon I realized that I could line up the hypotenuses of both triangles to create a rectangle.
Once the rectangle was drawn out I was faced with yet another challenge. I still only had one side length. I thought about the previous activities we had done, and came across the corral problem. By looking through my work from that assignment I was able to find the formula I needed, how to find the area of a rectangle with only one side length. I took 13, the one side length I knew and plugged it into the formula: x * 150 - x=Area. By doing this I got to an area of 1937 square feet. After that I used the area and the height to solve for the width. The width ended up being 149 feet. I then began to look at this as a 3D shape. I used the original circumference of the building as the length, since the slant is on the entire roof. At first I tried to calculate it as two triangular prisms, then I realized that calculating the volume as a single rectangular prism would be much easier. To do this I used the same formula that I used to find the volume of the whole building, and found a volume of 9,827,702.926555536 cubic feet. Now that I had the volume of the part missing from the roof and the whole building. To find the true volume of the building it took the volume of the missing part of the roof and subtracted it from the original volume of the building I had found. When I did 67,479,839.402789196 - 9,827,702.92555538 I got an answer of 57,652,136.47626405 cubic feet.
Overall I feel like I did a nice job on the design your own project. I managed my time nicely, and was able to pick an item that would be challenging for me. I feel like I could have thought it through a bit more and realized the building had nothing in the middle a bit earlier on, but other then that I am very proud of this project. I feel like the really helped put into perspective the sheer size of the building.
reflection
I believe this was a very strong project for me. I challenged myself sufficiently and was able to walk away with a good understanding of the material we focused on. I used the habit of a mathematician 'Take Apart and Put back Together' quite a lot on this project, especially on the design your own project. I broke down the strangely shaped roof of the building into shapes that I already knew. I also used the habit 'Start Small' on the final part of the corral problem, I began by finding the area of a 10 sided polygon, then moved onto a 33 sided polygon. I felt like that was a way to challenge myself, I noticed that something was too easy so I decided to crank it up a notch. During this project I did a much better job managing my time, I didn't find myself rushing as much to turn something in. I really enjoyed this project as a whole, I found it to be just challenging enough, I wasn't frustrated, and I wasn't bored. I enjoyed getting to see the real world application of what we were learning through the Everest Problem, and The Design Your Own Project. Out of everything I enjoyed the design your own project the most. I enjoyed getting to pick an object and measure that. I felt like it added a personal touch to the project. During this project I did a much better job with staying organized. It was much easier to keep track of then other projects, I made sure to label everything and write down all of my work. Overall I am very proud of the work I completed.