Problem Statement
What area of the land can the cow graze?
Process
Initially for this problem I thought OK if the cow goes left and hits that first corner he'll only have 90'left of rope allowance for to graze and if he keeps going he'll only reach 80', so on and so forth. Then me and my group realized that the cow will hit 80' and will want to go back and graze the other side of the barn instead of going all the way around and we and diagrams to go along with it (to the right)
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The diagram in the top right corner is a picture of our final diagram, and as you can see we split it up into smaller pieces so that it would be easier to find the area of the circle and also because it wasn't a full circle, we had many things that we had to take account for, for example the little dip the circle has in the upper left corner. The dimensions that we have to know to be able to solve this are that the barn is 10X10 and that the cow is tied up to the outside of the barn on a 100 foot rope. It's important to know the dimensions of the barn because you'll have to subtract it from the final answer since the cow is not grazing the inside of the barn. Knowing the length of the rope is also important because you have to know how much allowance the cow has to be able to move around and graze.
Solution
Step 1: We have to find the area of 3/4 of the circle. For this we used the formula A= π r^2 we plugged in 100ft for R because thats the radius of the circle and then we did 100^2 and that equals 10,000 then we multiplied that by π and we got 31,415 then multiply that by .75 because thats 3/4 but in decimals and you get the area of 3/4 of a circle which is 23,562
Step 2: We then had to find the base of the triangle. We used the formula A^2+B^2=C^2 for B&A we plugged in 10 and we left C alone because thats what we were trying to find. Then we did 10^2 for both A&B and you get 100 you the add 100 & 100 and that equals 200 then you do the square root of 200 which is 14.14 and thats the base of your triangle.
Step 3: Once we have found the base of the triangle we find the height of it using the formula A^2+B^2=C^2 to make things easier we could divide the triangle in two's the only thing we need to do is to divide 14.14 by 2 and you should get 7.07. Once you have that we could start finding the height of the triangle we plug in 7.07 for B and 90 for C, we don't plug in anything for A because thats what we are trying to find. Then you do 7.07^2 and that should give you 49.98 and you do the same for 90^2 and you get 8,100. Then you subtract 49.98 from both sides so 49.98-49.98=0 but 8,100-49.98=8,050 we do this so you could leave A^2 by itself then you √ 8,050 and you get 89.7 which is the height of the triangle.
Step 4: So we know have the measurements for base and height so we could figure out the area of the triangle and for that we use the formula A=B*H/2 for B we plug in 14.14 which is the base of the triangle and for H we plug in 89.7 which is the height of the triangle we then divide them by 2 and should get 634.179 and the we subtract 50 from that because we don't need that part of the triangle and we should get 594.2 and thats the area of the triangle.
Step 5: We know have to find the angels of the triangle. We already know that one of them is going to be 45 because of the barn but we still need find the other two angels and we use the formula SOHCAHTOA to find the angels. To find one of the angels we use SIN and the height of the triangle which 89.7 and we divide that by 90 so we do SIN^-1=(89.7/90) and that should give us 85.3 and that the other angels. We know have to find the other angle and what we do for that one is get 85.3 and add 45 that equals 130.3 and subtract 180 because all angels have to 180 and you get 49.7 and that the other missing angle.
Step 6: We know have to find the size of the triangles that are are next to the one that we were previously solving. For this we get 23,562 which is 3/4 of the circle and add 584.2 which is the area of the triangle and that equals 24,146.1 and the we subtract 31,172.3 and that equals 7,026.2 and divide that by 2 because there is 2 big triangles and that equals 3,513.1.
Step 7: The last step for this problem is to find the area of the circle where the cow will be able to be grazing through you add all of the past answers together and you should get 31,172.3.
Evolution + Reflection
I found this problem pretty hard because I felt like I did not understand anything that was going on half of the time but I really tried to have a good understanding of the problem and how to solve it. I also really enjoyed this problem because it was fun and it really tested everything we learned over the past two years (Freshman and Sophomore Year's) so it really tested my thinking and I enjoyed that the most. Something that I got out of this was getting to work with the new people in my class and getting to work with them, it was also really cool to be able to see everyones way of thinking and who I can go to when I need help. The group quiz was very beneficial to me because I got to put everything I learned and everything I already knew to test and I also got some help from my table mates when I was confused with something which was a +. For this problem I think that I deserve a B because I could still work on improving my skills on the subject, I also feel like I deserve a B because I tried my best to grasp the concept but I'm not quite there yet which is why I feel like I'm not at an A yet.