Problem Statement: There is a 10 foot by 10 foot barn, Daisy the cow is tied to one corner of the barn at the end of a 100 foot rope. What we want to figure out is how many square feet the cow can graze, while being attatched to its rope and going around the barn.
Problem Process: As you can see below in my first attempt of a diagram I didn't consider that the cow could go both directions or that Daisy would be able to walk almost all the way around the barn to graze. I also didn't get that the shape that the cow would be able to graze would resemble a circle.
Below is my final accurate diagram. As you can see it is more proportional and you can tell that it resembles a circle and that the point where the cow is tied to the barn is the center of the circle. You can also see how the rope gets shorter as the cow travels around the barn, this diagram also shows the area the cow can graze if it travels in both directions.
When we got the accurate diagram we knew what we needed to find the area of that weird shape, but there is no formula to solve for a circle with a dimple in it. We couldn't use the formula to solve for the area of the whole circle, because it wasn't a full circle. So we started by trying to cut up the large shape in to smaller pieces that we have formulas for so we can calculate it.
Doing this was useful because seeing 2 triangles 2 pizza slices and 3/4 of a circle was easier than seeing the dimpled circle shape, I felt less overwhelmed when we broke down the whole circle. The dimensions you need to solve this problem are the accurate measurements for each section, you know the uninterrupted length of the rope is 100ft but when it starts to go around the 10ft barn you know that the rope is then only 90ft. Which is important to know when drawing your diagram proportionally.
Problem Solution: So solving the cow problem is a big task. I broke it into 5 general steps. 1. Find the area of 3/4 of the triangle: A=(pi(r^2)).75 2. Find the area of the triangles: a^2+b^2-C^2 and A=h(b)/2 3. Subtract the 1/2 of the barn that is cutting into the triangle 4. Find the area of the sectors/pizza slices: theta/360(pi(r^2) 5. Add all the area together to find the total area that Daisy can graze But in more detail: the formula for finding the whole area of a circle is pi(r^2) and we multiplied it by .75 because we only wanted 3/4 of the area of the circle. A=23,561.94. This was the simplest part of solving the problem, when you get to the triangles is when it gets complicated.
When you look at the triangle you know the side lengths are 90 because of the 10ft of rope that gets cut off by the barn, the problem is you don't know the base or the height of the triangle which is what you need to solve for the area. But since the triangle is cutting into the barn there is the triangle that fills up the rest of the barn that we can use to solve for the base of the triangle we need. We use the Pythagorean Theorem (a^2+b^2=c^2) so c=14.14 so now we have a triangle with a base of 14.14 and 2 side lengths of 90. We still need to find the height of the triangle to solve for the area, so by cutting our triangle in 1/2 we can solve for the height using the Pythagorean Theorem again. h=89.7 and plugging the base and height into the formula to solve for the area, A=634.179. But the problem with this area calculation is that it includes 1/2 of the barn so we need to subtract the 1/2 of the barn from it. 634.179-50=584.2 so the area of the triangles is A=584.2.
Now all thats left is to find the area of the sectors or pizza slices and add it all together. What you need to solve for the sectors is the angle of the sector that we are solving for. We didn't have that, but we did have the 90 degree angle from the barn so we know that the line that goes right through the right angle is 45 degrees, we just have to solve for the other 2. Since we have the triangle that we needed to find the area of we can use that triangle to solve for the shared angle it has. We used SOHCAHTOA to solve for the missing angle. Theta=85.3 and since we know that and we know that the other angle was 45 we know that all 3 angles have to add up to 180 because of the straight line. 85+45=130 and 180-130=50 so the missing angle =50. Now that we know that we can use the formula to solve for a sector, which is theta/360 (pi(r^2). A-3,306 and we multiply that by 2 because there are 2 pizza slices so A=6,612.
Now to get the total area a cow can graze we add all 3 of our areas together. 23,561.9+584.2+6,612=30,758.1= the total area Daisy can graze.
This whole problem pushed my thinking, its mind blowing to think of how much I learned just from looking at my first attempt. The most challenging part for me was simplifying radicals I did not understand it at all but after practice and tutoring I feel much stronger with that concept. I think I got the most out of practicing and knowing all of those formulas to solve for the different areas, because now I can probable solve for any weird circlish shape by breaking it down. My group didn't really affect my learning, I think I was the only one that took time out of class to study and to practice to make sure I really had it so my group was helpful in that they challenged my thinking by asking me questions or to explain something but I feel like they didn't contribute as much as I did. If I were to grade myself I would give myself and A+ because I turned all of my work in on time, I completed most of the packets for practice, and I put a lot of time in outside of class to learn these concepts. Also I feel like I was steering my group in the right direction most of the time, like keeping them on track with discussions and I feel like I have a really good understanding of the concepts because my group would ask me questions.