The cow problem consisted of a barn that was 10X10 feet and a cow that was on a rope that was 100 ft long. That cow was attatched to one corner of the barn and we had to figure out how much land the cow can graze on. The picture above is the diagram I created after I cut up the shape which is something that happens in the first few steps to solving this problem.
The Process
My initial step that I took to try to solve this was draw out how the rope would go if it hit the corners of the barn and go around the whole thing. I quickly realized that the cow can go either way of the barn and would probably use both ways so that it can graze. Knowing that information we came up with the diagram that I showed in the "Problem Statement". We realized that it wouldn't be a perfect circle and that it would have some sort of dimple where the cow stops on both sides of the diagram. From there I knew we had to cut up this shape into smaller shapes that we could actually find the area for which is where we got the shapes from. We identified these shapes as arrows(two triangles), pacman(3/4ths of a circle) and "pizza slices"(triangles with a curved side). From here we went back to get practice on things that we had learned in the years before such as Trigonometry and the pythagorean theorem.
The Solutions
To Solve the "pacman" shape all we had to do was get the equation for the area of a circle which is PI*r^2, this then converted to PI*100*.75. I took 100 from the radius of the circle created by the rope and the .75 is there because it is only 75% of the circle. This then ended up equaling to 23,561.94
In order to solve the arrow like shape it was much more difficult and tidious than the circle. The first step was to add a part of area from the barn to make it an isosceles triangle and that way be able to find the area. I then did the pythagorean therium (a^2+b^2=c^2) and plugged in the numbers necessary (10^2+ 10^2=c^2). After solving that, as you can see in the image above, I ended up with 14.14 =c. I then moved on the cutting this triangle in half to be able to find the height. I did the pythagorean therium once again but with new numbers of course. After plugging in the numbers I got h^2+7.07^2=90^3 which then proceeded to be h^2+50=8,100, after solving this I got h=89.7. After that I solved for the area of the triangle using A=b*h/2 which after plugging in the numbers (in the image) gave me 634.18. I then subtracted 50 from this number because I needed to take away the amount of area we took earlier to make this an isosceles triangle. The final area is 584.2.
To figure out the area of the pizza slices we had to get the angles of each intersection. We already know one of the angles which is 45• because it is half of the angle we had when we took the area of the barn a couple of steps back. This angle is C, C=45•. The next angle I solved is angle b. I solved for this angle by using the inverse of sin. You plug in Sin^-1(89.7/90) to your calculator and you get 85.3 which is your angle. In order to find the last angle you add both angles you already have, 85.3+45=130.3 after that you subtract 130.3 from 180 since it's the total angle for half a circle. After you substract you get 49.7 which is angle A. After all of that, you divide 90/360 and get .13. You then use that .13 to find the area of the circle. Afterwards, you have to find the area of a circle which is PI*90^2*.13 which then comes out to a total of 3,306.42 which you then multiply by two because there is two pizza slices. This all gives you a total of 6,612.
At the end of figuring out each shape, you simply add them all together to get the total area. 6,612(from pizza slice)+584.2(arrow)+23,561.94(pacman)=30,658.14
Reflection
I really enjoyed this problem because it had so many different components to it that you really had to understand to be able to solve the whole problem. Once we realized how many parts there were, I noticed how many things I already knew but also how many things needed to learn. Some things were refreshers like trigonometry and the pythagorean theorem. Some things were things that I learned as we went (Inverse Trigonometry). I think that the group quiz did a lot for me because I was able to clarify my own understanding of the problem while helping my teams and they also helped me clarify things that I didn't understand 100%. I do ,though, feel like I did a lot or most of the work in my group during the quiz. I do think my whole group understood how to solve the problem, they just didn't contribute enough to the actual steps to get there. I think that I deserve an A because I knew most of the things that were going on during class and if I didn't I would ask clarifying questions. I also tried to help my groupmates understand what was going on as much as possible.