Progress
Initial Attempts:
after some discussion with my table we got here where we had to break up the grazing area we got on a conclusion that we had to separate our cirlce in 3 recognizable shapes that we knew how to get the area from to then add up all of the individual areas and get a general area of the whole figure, 1st shape is 3/4 of a circle , 2nd a triangle we split on2 pieces and 3rd some pizza slice looking shape that we knew it had the same area as the other one.
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Finding the individual Areas
the first shape. Like I said before, this shape is just 3/4 of a full circle. To find the area I first started off by finding the area of a full circle using the radius of 100 since the rope that's attached to the barn is 100 ft. long. My team and I plugged 100 into the equation A = π r² and got 31,415.9265. Since we're only trying to find 3/4 of the full circle, I multiplied 31,415.9265 by .75 which gave us the final answer of 23,561.9. wich was the area of the 3 quarters of circle
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Finding the area of this shape was a bit of a longer process with a lot of steps involved. To make solving for the area of this shape easier, we added a diagonal line right across the barn to create a complete triangle instead of working with the "weird arrow shape". Before jumping into solving for the area, we first needed to find the base of the triangle. Here's where it get's interesting, the base of the triangle also happens to be the hypothenous of the triangle right below that was created when we cut the barn diagonally in half. Therefore, if you find the length of the hypotenuse of that little triangle, you'd also be finding the base of the larger triangle. I found the length of the hypotenuse by using the equation A² + B² = C². We know that the barn is 10 x 10 and C² is always the hypotenuse, so I simply plugged in 10 for A² and B². I ended up getting14.14 as my answer which is the base of the triangle! Once I knew what the base of the triangle was, the next step was to find the area. To find the area we needed to first find the height of the triangle, to make this easier I cut the triangle in half making the base 7.7 which is half of 14.14. Now we can solve for the height once again by using A² + B² = C². I got 89.7 as the height of the triangle. Once I had all the base and height of the triangle, I was finally able to solve for the area by using the equation B x H / 2 . I got 634.179 as my answer, but this wasn't the final answer. remember how I added half of the square barn to turn the "weird arrow shape" into a complete triangle? Well now I need to subtract that half again because it is not part of the original shape. Since 10 x 10 = 100 we know that the area of the square is 100, and since I needed to subtract half of the full area of the square, all I did was subtract 50 (which is half of 100) from the answer. This makes the final area of the shape 584.179 or 584.2 (rounded). |
The final thing I had to do was find the area of these "pizza slices". I already knew the radius was 90, since the pizza slices start on the left side of the barn. If the barn is 10 ft. wide and the rope is 100 ft. long that means that when the cow walks to the left of the barn all alongside the edge, when he it gets to the left corner, it'll only have 90 ft. of rope left to move around with... in other words (100ft. - 10ft. = 90ft.). Since I already have that information, all I needed to do was find the angle of the "pizza slices". Doing so was kind of confusing to understand at first, but the idea is simple. Basically, the edge of the barn and the side of the "pizza slice" line up in a straight line. I basically drew half of a circle on the straight line because I knew that half a circle = 180°. By finding the 2 angles (on the diagram) I could then add them together and subtract it from 180 to find the angle of the "pizza slice".
The angle farthest to the right (on the diagram) I knew was 45° because it was half of a right angle, and a right angle is 90°. To find the angle in the center I needed to once again use the half triangle to find the missing angle. I did this by using SOHCAHTOA. we ended up getting 85° as the second angle. Now all I had to do was add 85° + 45°, which equaled 130°. Then, I subtracted 130° from 180° and I got 50° which was the angle of the "pizza slice" (angle farthest to the left)! After doing this, I then had to find what percentage of a full circle the "pizza slice" was by calculating 49.7° / 360° which gave me the answer of .138 0r approximately 13.8%. Lastly, I used the equation A = π r² to find the final area of the shape. π x 9² = 25,44.69 and 25,44.69 x .138 = 3,511.6. Since there are two of the exact same "pizza slices", I multiplied the answer times 2 which gave me the final area of 7,023.2! |
Final Step : Adding The Area
Reflection
while doing this problem was like going back in time and learned again everything I forgot in the past years this problem really pushed me to learn new things and how to put them together, it was frustating at the beginnig because I didnt knew what to do and I was completely lost but after practicing and working with my team I found the way to do it and i am very proud of that.