Problem Statement
This is problem we were given; There is a square barn that is 10x10ft. A cow is attached to one corner of the barn on a rope that is 100ft long. What is the area that the cow can graze?
We started by drawing a diagram that helped us visualize the area the cow can graze.
We started by drawing a diagram that helped us visualize the area the cow can graze.
If we imagine the cow is attached to the bottom right corner, we can see the ways the cow can easily wrap around 3/4 of the barn. In the top left fourth of the circle, there is a dip. This dip represents the spot where the cow cannot reach. This dip will be apparent, even if you increase the barn size or decrease the rope length.
Our task was to find the area of the whole circle, including the area with the dip.
Process
To solve this problem, we were given packets that helped us hone the skills we needed to solve the cow problem
After becoming familiar with the math concepts necessary to complete the cow problem, we solved it as a class. In the slideshow, you can follow the process I used to solve this problem. In the first picture, something to note is the colors. As a class we decided to section off the top left fourth of the circle into triangles because calculating the area of a triangle is much easier than calculating the area of an odd shape.
Solution
To find the solution to the cow problem, we had to find the area of all of the shapes then add the areas together. Simple enough right? That is what I thought, but the actual process is a lot more complicated. Let's walk through it together.
First, let's solve for the area of the largest part of the circle (If you refer back to my original diagram, its the part outlined in green). We know the formula to find the area of a circle is pi times radius squared, so all we need to do is plug in the values we know. The radius of our circle is 100 because the length of the rope the cow is attached to is 100ft long. Then, we can multiply 3.14 (pi rounded) by 10,000. This leaves us with 31,400 BUT this is not the area of 3/4ths of the circle. This is the area of the whole circle. To find the area of 3/4ths of the circle, we multiply 31,400 by .75 (3/4ths as a decimal), and we are left with 23,550, which is the area.
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Next, we need to find the area of the triangle that intersects with the barn (in my diagram it is outlined in yellow). To start, let's use the Pythagorean theorem (a^2+b^2=c^2) to solve for the hypotenuse, or the longest side of the triangle inside the barn. This will be our base. Plug in the side lengths of 10 to a and b, then solve using basic algebra. To solve for the area of a triangle we need the base and the height of the triangle, because the formula is base times height divided by two.
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The second step to finding the area of the yellow triangle is to find the height of the triangle. We can use the Pythagorean theorem again, first cutting the triangle in half. Plug in 7.07 for b, and 90 for c, and lets solve for a (or h for height). Solve, like you normally would. We get 89.7 as the height of the triangle, and now we have all of the values we need to find the area.
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The formula to solve for the area of a triangle is base times height divided by two, and since all of the hard work is done, all we need to do is plug the values in! 14.14 is the base, 89.7 is the height, and when you multiply them together and divide by two, you should get 634.2. Now, because we solved for the whole triangle's area we have to account for the section that is overlapping the barn, this is where the 50 comes from (The total area of the barn is 100ft, and half of 100 is 50). Now, you just subtract 50 from 634.2, and you are left with 584.2.
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The next step is to find the area of the two "Pizza slice" sectors (outlined in blue). First, we have to find the angle of the section outlined in orange, so that we can find the angle connected to the pizza slice. First, here is what we know. We know that angle 3 is 45 degrees, and that angles 1,2,and 3 have to add up to equal 180 degrees. We can use SOHCAHTOA to find theta (the missing angle). Let's label the sides of the triangle. The hypotenuse is 90, the base is 7.07, and the opposite is 89.7, all figures we have previously solved for. Using sin -1 we can solve for the angle by plugging in opposite over adjacent (base), giving us 85.3 degrees. We can assume that angle 1 is going to be 49.7 degrees because that is the only value that can be added to 45 and 85.3 degrees, to equal 180.
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Finally, to solve for the area of the pizza slices, we take 49.7 degrees and divide it by 360 because 360 degrees is a circle. We do this to find how much of the circle the pizza slices take up. We get .138, or around 13.8%. Lets find the area using the basic formula, pi times radius squared. We know the radius is 90, this leaves us with 8100pi, we can multiply by pi to get 25,446.9. Last but not least, we have to multiply 25,446.9 by .138, which leaves us with 3513.1. This is the area for only ONE of the pizza slices, but all we have to do is multiply 3513.1 by two, to find the total area.
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Evaluation/ Reflection
The aspect of the problem that really pushed my thinking was the SOHCAHTOA and radicals aspect. I learned trig in freshman year but I did not remember anything from it going into the unit. I also have not had much experience with radical numbers. I was not able to really grasp SOHCAHTOA or anything to do with radicals until the day before the cow problem exam, when I reached out to my peers for help. They were able to explain it to me in simpler terms, and work with me step by step.
The part of this unit that I got the most out of, was when the whole class worked on solving the cow problem. Before we solved the problem, I was completely lost. I had no idea how trigonometry applied to the area of a triangle, but when Mr.B explained how to find the area of the different sectors it all clicked.
The group quiz was a bit of a dramatic experience for me. I was placed in a group with people who I have worked with previously and based on this prior knowledge, I knew I needed to step my game up because I was the only hope for the group to at least slightly succeed. The Wednesday before the group quiz, I was frantic to find help on the aspects of the cow problem that I had no idea how to do. I turned to my peers and they guided me through the confusion. Using the packet that was given as a study guide, I answered at least one question from each section with a thorough process in the packet. At home, I completed the packet. Looking back, I know that this was not enough. I wish I had went to tutoring or spent some time on Khan Academy. I did not do as well as I should have on this exam, and this experience serves as a lesson in the future. That lesson being, I NEED TO STUDY MORE.
If I were to give myself a grade on this unit, I would honestly give myself a B. This unit was more complex than I thought it would be and I did not devote enough time to studying and really understanding the strategies we were using to solve the cow problem. Here are examples of why I would give myself this grade.
The part of this unit that I got the most out of, was when the whole class worked on solving the cow problem. Before we solved the problem, I was completely lost. I had no idea how trigonometry applied to the area of a triangle, but when Mr.B explained how to find the area of the different sectors it all clicked.
The group quiz was a bit of a dramatic experience for me. I was placed in a group with people who I have worked with previously and based on this prior knowledge, I knew I needed to step my game up because I was the only hope for the group to at least slightly succeed. The Wednesday before the group quiz, I was frantic to find help on the aspects of the cow problem that I had no idea how to do. I turned to my peers and they guided me through the confusion. Using the packet that was given as a study guide, I answered at least one question from each section with a thorough process in the packet. At home, I completed the packet. Looking back, I know that this was not enough. I wish I had went to tutoring or spent some time on Khan Academy. I did not do as well as I should have on this exam, and this experience serves as a lesson in the future. That lesson being, I NEED TO STUDY MORE.
If I were to give myself a grade on this unit, I would honestly give myself a B. This unit was more complex than I thought it would be and I did not devote enough time to studying and really understanding the strategies we were using to solve the cow problem. Here are examples of why I would give myself this grade.