Cow Problem
Process/Solution: My initial attempt to this problem was to draw out and try to better understand what I thought it would look like.
These two bottom drawing are what I sketched out in the beginning. The first one better represents the official diagram we would use in class.
These two bottom drawing are what I sketched out in the beginning. The first one better represents the official diagram we would use in class.
The next thing I did was try to figure out the area we could already get. From previous math, we knew how to find the area of a circle. To find the area of a circle sector, we found out that you would multiply the whole circle area. Here, we have 3/4 of the circle and we would multiply that by the original area of the whole circle to figure out the sector's area. Below is the work for that.
I wanted to find the area of the new triangle I had drawn, but I was stuck because we need two things to find the area of a triangle: the base and the height, two things we did not have. We did however, have the ability to solve for the base and height using Pythagorean Theorem. |
The area for 3/4 of the circle was about 23,550 feet^2.
My next step was to calculate what was left of the area. I started adding onto my diagram and realized that from the opposite corner of the barn from where I started off with, the distance was similar to 90 feet long. |
First, I found the hypotenuse of the 10 by 10 barn, when split in half.
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Using that, we could solve for the missing side of the bigger triangle.
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Now, we could find the area of a triangle because we found the base and the height. Once I found the area of the whole triangle, I subtracted 50 from the total because that part was part of the barn. The area of the triangle was about 584.2 ft^2.
Our last step was to calculate the area of the missing sectors.
To do this, we have to know how to use Sin, Cos, and Tan, but better yet, SOHCAHTOA. Normally, we would use sohcahtoa to find a missing side length. However, what we wanted to do was find the angle for x.
To do this, we have to know how to use Sin, Cos, and Tan, but better yet, SOHCAHTOA. Normally, we would use sohcahtoa to find a missing side length. However, what we wanted to do was find the angle for x.
Once we knew x, we could use our knowledge of triangles and circles to figure out the last missing angle. Since we know that circles equal 360 degrees and half of a circle equals 180 degrees, we could figure out the last missing sector area. At first, I added x to the angle we already knew which was 45 degrees. From this, we could subtract 13.03 from 180 to find the missing angle.
Using the degrees, we were able to find the area of the missing sector. Like earlier, we took the degrees and found the percent of the circle it took up. To the left, is how the area of a single sector was found. Then, we added another sector because there were two of them. The area for both sectors is about 6612.84 feet^2.
The last step is to simply add all of the areas together All of the areas added together are 30747.04 feet^2. |
Reflection: I really enjoyed doing this problem because it had a lot of steps that required me to bring concepts that I have learned in the past, all together. Something that really pushed my thinking were the inverse functions. Previously, I had not worked much with trigonometry and Sin, Cos, or Tan so this was a little difficult to pick up at the pace we were going with the problem. I think that having the group quiz really helped me understand the problem because we had been working with our group and we all learned how to cooperate with one another. It was also good to catch simple mistakes that would change the entire solution of the problem. During the quiz, we had it all written down, but one of the sides were misrepresented (hypotenuse, adjacent, opposite). It was extremely helpful to have three extra pairs of eyes because we were able to catch each other's mistakes. If I were to grade myself on this unit, I would give myself an A. I believe that I worked well with my group, I was always on task, and was able to follow along with the process as well as ask questions throughout it. I was able to complete many worksheets that would better help me understand the concepts and push my thinking.