## The Cow Problem

A farmer ties his cow to one end of his 10' x 10' square barn with a 100' rope. What we need to figure out is the maximum area in which the cow can eat the grass around the barn.

## Process

Together we came up with the image to the right. Although a few of the measurements are off, the "circle's" shape clearly shows the general area in which the cow is able to move within and what shapes we had to solve for.
By breaking up 1/4th of the circle that had an awkward shape to it with manageable pieces like triangles and partial circles made solving for the entire area much easier. Essentially, solving for the area of a triangle is much easier than having to come up with some type of formula to calculate the area of a circle that is missing a very small percentage of itself. |
When we were first given this problem we were asked to provide some sort of visualization of what we were solving before trying to actually solve for something.
Independently I came up with the larger circle you can see on the left. Based on the notes and what I saw others sketching, I proposed this image to my group. As soon as I saw what they had came up with I recognized that I was wrong. |

## Solution

After calculating everything necessary in order to find the entire area in which the cow could roam around the barn we found that that area was 31,701.13ft^2.
In order to come to this solution we found the area of one triangle and a portion of a circle (what is highlighted in yellow). After finding the area of these two figures, we added them together. After this we multiplied the sum by 2 (because these two shapes mirror what is parallel to it). Then we calculated the area of half of the barn and subtracted that to our entire area. Once we found the area of 1/4th of the entire shape, we added it to 3/4th of the rest of the shape. The formula we used looked like this; A = A3/4 of the circle + (A of the △ + A of the Sector)2 - 1/2 of the barn Finding the area of these manageable shapes took many steps. We learned how to use trigonometry, how to find the area of a partial circle, how to find the area of basic shapes like triangles and squares, and learned how to simplify the square root of a number. |

## Reflection

Since we first began the problem I was in the mindset of trying my best to understand the problem and the steps that went behind finding the solution. I had a difficult time understanding trigonometry and how it was going to be used in this problem. The concepts and the notes I took on the subject did not make sense to me, let alone did applying them to the Cow Problem.

As a result I reached out to my group and asked for the math to be worked through with me and for the concepts to be explained. After doing this and practicing it on my own at home (as you can see from one of the problems from my worksheet on the left) I was able to explain to other people in the class and show how it was used in our problem.
From this experience I learned that no matter how scary a problem may initially look, with dedication and hard work I can solve anything that is presented to me in the upcoming units. |

The closing of this unit was taking an independent and group quiz. The group quiz meant everyone in my group of four having to understand all the components of solving the problem and contributing their ideas on how to solve the problem that was similar to the Cow Problem. Luckily, I was fortunate enough to have a group that was engaged and wanted to get a good grade of this test.

Two days before the test we went over all the concepts and skills necessary in order to solve the Cow Problem. Doing this had everyone become an active member of the group, wether that meant asking questions or answering them, everyone made sure that at the end of the two days they understood the problem. Personally, this practice bettered my understanding of the problem and made me feel more confident in my ability to explain difficult math.

If I were to grade myself on this unit I would give myself an A. Throughout the lessons and practices I have been an engaged student in the classroom. I was always willing to help others and made sure that I understood the problems we solved.

Two days before the test we went over all the concepts and skills necessary in order to solve the Cow Problem. Doing this had everyone become an active member of the group, wether that meant asking questions or answering them, everyone made sure that at the end of the two days they understood the problem. Personally, this practice bettered my understanding of the problem and made me feel more confident in my ability to explain difficult math.

If I were to grade myself on this unit I would give myself an A. Throughout the lessons and practices I have been an engaged student in the classroom. I was always willing to help others and made sure that I understood the problems we solved.