Before The Project
Before we went out and actually measured anything we Spent 2-3 weeks exploring equations of lines, area, volume and how they all relate to each other. We started with Pythagorean Theorem witch gave us with the base line for our studies into Trigonometry. After covering both the uses and meaning of Sine, Cosine, and Tangent. We moved onto the distance formula and stated to look at Dimensions: 0D equaling a single point. 1D dragging that point creating length, 2D dragging that line up and creating area. And dragging that plane up creating the third dimension creating volume. We found the relationships between different area formulas, (why we use pi for circles and how that relates to squares), And how base times height relates to all volume formulas. One of the foundations of our outstanding on why for example pyramids are equal to 1/3 of the area of a cube with the same base and height. The reason was because of cavalieri's principle, which states that if a shape is slanted but retains it's height and base the area remains the same. What that means is that if you have for example, a stack of coins and you slant them to the side, their are the same amount of coins. This helped with specifically with cubes and their relation to pyramids. For that relationship you create three identical pyramids and, because of cavalieri's principle we can shift the piont or height of the triangle to the center creating a traditional pyramid. This understanding was critical to learning other relationships and how we derive area and volume equations from them.
The Relationships
Starting with Pythagoras Theorem of A2+B2=C2 is equal to A2+B2=C or the distance between A and B Replace A with (x1-x2) and B with (y1-y2) and you get
Distance =(x1-x2)2+(y1-y2)2
After going over how Sine(x)=Opposite over Hypotenuse Cosine(x)=Adjacent over Hypotenuse, and Tangent=Opposite over Adjacent side lengths, we went on to area. We covered how as equilateral shapes have more and more sides and their parameter are equal, their area will get closer and closer to (Pi)R2 Times Height. And how Volume of many shapes are equal to base times height including cylinders.
Distance =(x1-x2)2+(y1-y2)2
After going over how Sine(x)=Opposite over Hypotenuse Cosine(x)=Adjacent over Hypotenuse, and Tangent=Opposite over Adjacent side lengths, we went on to area. We covered how as equilateral shapes have more and more sides and their parameter are equal, their area will get closer and closer to (Pi)R2 Times Height. And how Volume of many shapes are equal to base times height including cylinders.
The Project
At the very beginning of the project we were tasked with coming up with our own project. The only requirement was that we would have to calculate the length, area, or width of a certain object. It was up to use to both find something interesting, and also challenge ourselves. We could choose to be in groups of up to three people, but I choose to do it myself. We then were tasked with creating a presentation showing a brief summary, the math behind it, and a short reflection. The final step was putting it all up on one DP Update.
What I Measured
NRG Cabrillo Power plant’s smoke stackSummaryI measured, the NRG Cabrillo Power plant’s smoke stack through the use of trigonometry, ground measurements, and google maps. I used google maps to find the diameter of the smoke stack’s exterior and Interior. As well as the the distance between a certain point and the smoke stack. From that point I measured the angle from the base of a point to the top of the tower. From there I found the height of the smoke. Next I found the area of both the interior and exterior cylinders. Finally subtracting both cylinders volume gave me the total volume of the smoke stack.
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Diameter of Smokestack |
Height Part 1: Length |
Height Part 2
Now that I had the length and angle (44.5) I could use the Trigonometry equation for Tangent that the tangent of an angle was equal to the Opposite (Height) over the Adjacent (Length). So H/403.225865Ft=Tan(44.5). Multiply 403.225865Ft by both sides and you have H=Tan(44.5)*403.225865Ft and after calculations that means that the height is equal to 396.248954Ft
Calculating Volume Part 1With The height and the diameter it was then possible to calculate the volume of both cylinders using the equations below.
H=396.248954Ft Volume of B=H*Pi(41.30434Ft/2)Squared Volume of B=536944.8152Ft Cubed Volume of A=H*Pi(32.60869Ft/2)Squared Volume of A=330820.0225Ft Cubed |
Calculating Volume Part 2The Final step is the most simple. It just involves subtracting the Volume of Cylinder A from Cylinder B, equations below:
VolumeB- VolumeA 536944.8152Ft3-330820.0225Ft3 Actual Volume: 200124.7927Ft3 |
Reflection
Personally I really enjoyed this project. Even if the match itself wasn't that challenging, I feel like that was the point. The point was to figure out how to actually measure something even if you know little to none of it's dimensions. It was about using the tools we were given in class, and apply it to the problem at hand. Forcing us to see how the topics we learn in class can actually apply to the real world. In my problem that was my favorite part being able to go out and actual measure and use tools like google maps to find heights and length of objects I see fairly often. The hardest thing was actually going out to the location and measuring the angle for the height as everything relied on that part. But even with that challenge I did get as close to the correct height as I could have, which made me personally very proud. When solving the problem the most important thing I did that was so important was taking apart the problem into smaller tasks. Given the fairly short amount of time given it was important to know what exactly I was doing. So splitting the problem up into 4 parts was incredibly helpful both for reducing stress and managing time. In summary, while none of the maths were new, I did see a new way of applying the math we are learning in class to the outside world. I hope that in the future that we are given similar projects in structure.