## Problem Statement

In the cow problem, we had 10 x 10 ft. barn and a cow was tied on the outside, bottom right corner to the barn. The rope that tied him to the barn was a 100 ft. long rope. Let's say the cow's name is Bessie, it's a female cow. Bessie is trying to see how much land she can graze around her barn, still being tied up. She is a smart cow, so she is going to graze the most land as possible, we just need to figure out how much land that is.

## Process

My first attempt at it, looked like this

This being my first attempt, I tried to calculate how much rope Bessie would have when going around corners and certain spots. What I figured out in the second sketch ( the one that is 3/4 of a circle) is that, from those two cornes it can extend without shaping the rope into a different angle. Overall, this attempt, especially the first sketch(top left) was not very accurate at all, especially because I accidentally started from the top right corner and from where I drew the the lines, I didn't use the rope to properly and I ended up with an incorrect depiction of what the best way was to graze the most area.

After my first attempt being very inaccurate, I knew I had to better myself, so I focused a lot on solving practice problems and understanding each concept! There is more to just this page, there are like 3 more pages of different concepts, involving the Pythagorean theorem, sochahtoa, areas of cirlces, triangles, etc. This helped me in understanding what steps I will encounter during the process of a solution!

Here are some crucially important concepts I had to have mastered:

Here are some crucially important concepts I had to have mastered:

**Area of a circle: πr²**

Area of a triangle: bh/2

Area of a square= L x W

Phythagorean Theorem: a²+b²=c²

SOHCAHTOAArea of a triangle: bh/2

Area of a square= L x W

Phythagorean Theorem: a²+b²=c²

SOHCAHTOA

## Breakdown of Final Solution

Step one: Draw out 10ft x 10ft barn with the rope of 100ft extending to it's maximum capacity.

Step 2: Draw out the area I knew which was 3/4 of a circle. Then label the dimensions correctly. I had to figure out from what point, is what length. The rope becomes 90ft. when crossing the two first corners because the barn is taking up that length.

Step 3: Then we saw how if Bessie is going to graze the most land, she isn't going to cross a second corner, after she crosses one. The second corner in this case was always the top left. The reason why she would never cross it was because her rope will decrease even more and not allow to graze land that she can graze by simply going the other way. If she goes back and forth, crossing the top right and stopping directly above the top left, then going back around crossing the bottom left and going directly above the top left again, she grazes the most land. We figured out the length would still stay 90ft. when going above the top left because she isn't' crossing another corner. This created a little "dimple", in the our circle. we can't just solve the area as if it was a normal circle because it isn't one, so we needed to figure out another way.

Step 4: You have to break the "dimple", area into shapes you know how to solve for. We saw that there was a possible triangle that we could solve for to give us an area of a piece we need to solve, to solve other pieces ! When making the triangle we had to cut into the barn, and we calculated the dimensions of the triangle inside the barn using its 10ft. x 10ft and then calculate the hypotenuse.

Since we found the hypotenuse in the last step, we can use that now as a base for a new shape we can solve. First we need to find the height of the new triangle(the one connecting to the dimple) we can solve for. We break that one big triangle into two halves that now become right triangles. 10√2 now gets broken into 2 5√2's. Now we solve for the height using the two lengths we have which are 90 and 5√2 into the Pythagorean Theorem.

Lets look at the angles really quick. We can find the angle that is in the baseball field looking shape because we can find the angles in the other shapes neighboring it. We need to find the angle in order to find the area of it, so we find it by using our knowledge about the other angles we have connected to it. We know that it's a straight line so all three of those angles are going to add up to 180 degrees.

.Since we are nearing the end, lets calculate very quickly the space we know we can calculate which is 3/4 of a circle. We follow the steps of finding the area of a circle expect we need to multiply our result by 3/4 since we only have 3/4 of a circle.

Now back to the baseball field looking shape. We have the angle now and we know the two lengths around it so now we can calculate the area for it.

Looking back at the triangle we split into two , to make two right triangles, we can now calculate the area using base x height divided by two. That'll give us the area of that full triangle, but then we need to minus half the barn.

Then we add the area of the triangle, the baseball field looking shapes, and the 3/4 of the cirlce and we have the total area Bessie will graze if she grazes as much as she can.

## Mathematical Techniques

Like I listed above, here are essential techniques we learned in order to solve a complex problem such as this problem,

**Area of a circle: πr²**

Area of a triangle: bh/2

Area of a square= L x W

Pythagorean Theorem: a²+b²=c²

SOHCAHTOAArea of a triangle: bh/2

Area of a square= L x W

Pythagorean Theorem: a²+b²=c²

SOHCAHTOA

## Individual Quiz

Here is the individual quiz we took based on the concepts we learned in this problem. I got a score of 100/100 on it. Ignore the hand-writing, I know I need to improve it.

## Group Quiz

Here is the group quiz we took based on the actual cow problem, just with different dimensions regarding the barn and the rope. I also got a full score on it. (Disregard hand-writing once again.)

## Evaluation/Reflection

I think this problem overall pushed my thinking a lot. I learned a lot of these techniques in past years of math and online learner, but I am not the fastest learner when it comes to math. I forget formulas/methods quickly and in this project it sort of refreshed my memory, but overall this learning experience was very helpful because I remembered old concepts and used them in more difficult experience.

Taking the group quiz was good for me because like I said I wasn't very good at remembering concepts, but learning them and then vocally explaining it to my group at times helps me implant concepts better. It really had a positive affect on understanding concepts and that ultimately boosted my confidence in math.

If I were to grade myself on this unit, I would give myself an A because I came in trying to tackle these math techniques head on and would always ask my group members if I was lost at any point in the methods. I also took time after school to watch videos and do practices on my own on sheets of paper to understand SOHCAHTOA, Pythagorean Theorem, etc. I also contributed a lot in my group when it came to the group quiz, I felt like I was very vocal and clear when explaining steps, and answers, and came out of it with a full score. I also completed the individual quiz with the result being a full score again. I really feel like I became a vocal leader in this project when it came to asking questions or explaining something.

Taking the group quiz was good for me because like I said I wasn't very good at remembering concepts, but learning them and then vocally explaining it to my group at times helps me implant concepts better. It really had a positive affect on understanding concepts and that ultimately boosted my confidence in math.

If I were to grade myself on this unit, I would give myself an A because I came in trying to tackle these math techniques head on and would always ask my group members if I was lost at any point in the methods. I also took time after school to watch videos and do practices on my own on sheets of paper to understand SOHCAHTOA, Pythagorean Theorem, etc. I also contributed a lot in my group when it came to the group quiz, I felt like I was very vocal and clear when explaining steps, and answers, and came out of it with a full score. I also completed the individual quiz with the result being a full score again. I really feel like I became a vocal leader in this project when it came to asking questions or explaining something.