For this project, we measured every day objects using the Pythagorean Theorem, the distance formula, the equation of a circle, the unit circle, the definition of sine, cosine and tangent, right-triangle trigonometry, area of polygons, area of a circle, and then volume. On the side, there's a slideshow of the different equations for all of this content we learned.
To show a few examples of the equations we learned, I'm going to use the equations for the Pythagorean Theorem and the definition of sine, cosine, and tangent. For the Pythagorean Theorem, the equation is A^2+B^2=C^2. For the definition of sine, cosine, and tangent, the most known equation is SOH CAH TOA. SOH: Sine = O/H CAH: Cos = A/H TOA: Tan = O/A O = Opposite A = Adjacent H = Hypotenuse |
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To find the volume of something, you need to do the base of the object multiplied by the height. The only issue with this is that we had the base, but not the height. It wasn't like we could use a really long ruler and measure it out.
We knew that the flagpole itself was a cylindrical shape, so we measured the circumference of the flagpole to get the diameter and then the radius. The radius turned out to be 3 inches. Using the equation to get the area of a circle (pi*r^2), we found out the base of the flagpole was 28.26 inches. Now that we had the base of the flagpole, we needed the height. Personally, this part of the project was the most entertaining and also the most frustrating part. To get the height of the flagpole, we needed to use this really long measuring tape, a protractor, string, and a ruler. We used the measuring tape to measure out 30 feet away from the flagpole itself. That's where the protractor, string, and ruler would come into play. We used the ruler to measure out 2 feet and 3 inches above the ground. Then we used the string and protractor to figure out the angle of the flagpole compared to the spot we were sitting. The angle first turned out to be 55 degrees. We had plugged in the 55 degrees to this equation: 30 tan (55), and ended up getting 70 feet tall for our height. When we checked with Dr. Drew, he told us he seriously doubted the flagpole was 70 feet tall and was actually around 30 feet. Of course we argued that the angle we got was correct, so after a lot of arguing, we measured it out a second time. When we got the same angle, we were really confused. We took out the long measuring tool again and measured out 70 feet. When we saw it, it didn't look right, so we guessed the flagpole was more around 50 feet tall. When we looked at our angle again, we realized our problem. 55 degrees was the wrong angle after all. We had accidentally flipped the angles we needed, and to fix that, all we did was do 90 degrees minus 55 degrees to get 35 degrees. After plugging 35 degrees into our equation (30 tan (35)), we had finally gotten the correct height. The correct height of the flagpole is 48.01, which is around what we thought it was going to be. Using the new height, we plugged it into the volume formula to get 1,357.17 for the volume of the flagpole. |