Introduction & Problem Statement
This is Bessie the cow...
..She was tied on a corner on a square fence that had the length of 10 ft. and the width of 10 ft. She's a smart cow though, so she will not over cross with the rope and get herself tangled up. How much area of grass can Bessie eat?
Reviewing information we know... -The rope is 100ft. -The length and width of the barn is 10 ft. but we don't want to calculate the area of the barn with the area of the grass. In other words, we want to calculate the outside area of the grass that is outside the barn. |
Process Of The Cow Problem
1a.
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The first time I heard of this problem, I was confused and conflicted about the rules, expectations, and information because I felt like I was thrown into an arena full of lions without a sword. I didn't think I was going to answer the problem nor understand it. In picture (1a)., was the first time we were introduced to the cow problem. Instead of answering mathematically in the warm up time, I asked questions of the misunderstandings I seemed to trip over. |
Now we separated the diagram into manageable parts knowing what we could do with the formulas provided for us.(2a)
Now that we split it into somewhat circles and triangles, we first have to deal with with the unusual point or slit that made this diagram look like a heart. Since this triangle is a isosceles triangle, we could calculate one of triangles areas and multiply it by two in the end to make our lives easier. We could, only if we can meet the formulas requirements... 1/2BxH We first have to calculate the height of the triangle and the base of this triangle (2b). But how? Using half of the barn and it's sides are 10ft. in length.(2c.) This helps to figure out the base for the half of the triangle! Then, by using the Pythagorean Theorem to calculate the base. 10^2+10^2= c^2 100+100=c^2 200=c^2 Now we square root both sides to get rid of the confusing exponent.. √200=√c^2 √200= 10√2 c=10√2 We figured out the whole length of both triangles. In this case, I want only one triangle to not struggle further... 10√2/2= 5√2 B=5√2 |
2b. 2c.
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3a.
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Now that we have the base of one of the triangle we can now figure out the height to complete all necessary information to calculate half of the triangle's that is including the barn.
Still using the Pyth. Theorem... a^2+b^2=c^2 The hypothenuse is always c^2=H 90^2 + (5√2)^2=c^2 8,100 + 50 = c^2 8,150 = c^2 Square root both sides to get rid of that pesky exponent.. √8,150=√c^2 89.72 ft. = H Now we use the 1/2 BxH to calculate the area (1/2 5√2 x 89.72)= 317.21 x 2= 634.42 ft. What about the area of the barn that is included with the two isosceles triangles? You find the are using the information you received.. 1/2 10√2x10= 70.71 70.71-634.42= 563.71 |
Now that we figured out the triangle area(3b.), we have to solve for all these weird circles that are not as a whole.
Using the formula for finding angles in order to solve for a circle, I solved for this cut off circle. By using the side lengths of SOH-CAH-TOA to find theta angle, I had this formula. I decited to use sin because of the square root that I can avoid. sin-1(89.72/90)= 85.47 degrees (3c.)Then, I knew that half of 90 is 45, the angle next to it is 85.47 and adding those and subtracting the answer by 180 = 49.53 3c.
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3b.
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Now you must be thinking, why the heck did you spend all this time to answer for some silly angles! Fear not reader, because you're in for a little bit of magic. :)
When I was sitting in class puzzled about how my brain going to calculate 3/4's of this circle, our teacher, pulled out a erasable marker and show us that if you find out the angle of the circle and divide it by 360 and finally using the πr^2 formula you can calculate parts of a circle!
When I was sitting in class puzzled about how my brain going to calculate 3/4's of this circle, our teacher, pulled out a erasable marker and show us that if you find out the angle of the circle and divide it by 360 and finally using the πr^2 formula you can calculate parts of a circle!
Final formula of how you can calculate the are of a part of a circle by using the angle over 360 degrees
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Even if you have any doubts about this formula look at this! 90 degrees is the angle the partial circle is in. Divide that by 360 and you get 1/4! The obvious portion of the circle.
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Another example of the calculations due to the wonderful formula. Showing you that it gets you the exact fraction area of the circle you're answering for.
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4c.
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Now we needed to calculate the whole 3/4's of this circle of this diagram. (4c.) Instead of using the theta/360 method, I just used the fact that this portion was 3/4's of a circle and used that to multiply to πr^2.
We know.. -100 is the radius -the circle with 100 radius is 3/4's of this problem 3/4 x π 100^2 = 23,561.94 (4d.) Now all we gotta do is add em' up! 3501.07 x 2 + 563.71 + 23,561.94= 31,127.79 area worth of grass! |
4d.
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Reflection
What pushed my thinking mostly is my group members. Sarina especially, she provided a balance like a cane to keep me mentally stable when it comes to understanding the problem. Although, she helped me in the fact that she pushed me to my limits of understanding. I also personally helped others understand, but I would always refer to what Sarina would teach me. I got most out of the learning experience and the confusion and lack of motivation of math class. Finishing this problem was probably the most burdening yet most satisfying math question to finish. I never been this confused in a math class in a long time! I barely understood the exit card and after that day, I felt really bad about my performance in math class.
The group quiz is probably the most confident test I had in all my years of High Tech. I had this sense of trust and optimism because of one person. It was a positive experience because I think all of us had a major learning experience in the subject and as participating as a group member. If I could grade myself on this whole unit I would give myself an A, but for honest purposes though, I worked my butt cheeks off since I struggled immensely but reached a full understanding of the unit in the end. I never have grown so well in a subject in my life!
The work I did from the past to further my understanding...