**Problem Statement:**

There is a barn that is 10x10. There is a cow attached to one corner of the barn with a 100ft rope. We had to find how much of the area the cow could graze.

**Process:**

My first attempt to draw the diagram was just a square in the middle and a big circle surrounding it. At first, it seemed right, but then my group realized that the cow couldn't go around in a full circle because the rope was going to cut off. We knew it was going to be a weird circle, but we weren't really sure of how it was going to look. We decided to draw it out with the whole class, and this is what we got:

After finishing this diagram up, we knew something was off. What about the other part of the circle that was missing? We were stuck and didn't really know how to draw it out, until the class came together and drew it. It looked like this when it was done:

After finishing this diagram up, we knew something was off. What about the other part of the circle that was missing? We were stuck and didn't really know how to draw it out, until the class came together and drew it. It looked like this when it was done:

After drawing it, we knew we needed to cut this up into pieces that we knew how to solve the area for. We cut the circle in 3/4, then formed two triangles, and finally, what we call, the "pizza slices."

**Solution:**

We knew this was correct because the rope was getting shorter and shorter. We labeled it "turning point."

After this, we needed to break it down into pieces we needed to break it down into pieces we already knew how to solve for. For example, half a circle, or in this case, 3/4 of a circle. The formula for it was

Once we plugged it in and solved for it, the area for the 3/4 of the circle was 7,500. We had to do, 7,500*Pi = 23,561.9

After solving for this, we found more shapes that we knew how to solve for. For example, triangles. We cut out two triangles. At the end, our diagram looked like this:

**π r^2(.75)**Once we plugged it in and solved for it, the area for the 3/4 of the circle was 7,500. We had to do, 7,500*Pi = 23,561.9

After solving for this, we found more shapes that we knew how to solve for. For example, triangles. We cut out two triangles. At the end, our diagram looked like this:

In order to find the area of those two triangles, we had to find the height first. We solved for it by using the pythagorean theorem.

Using this, we knew the height of the triangles was 89.7. Then we needed to find the base, so again we used the Pythagorean theorem. For the base, we got 14.14.

Finally, to find the area of the triangles, we were able to use A=b*h/2

This step was easy, we just had to plug the two answers in.

This step was easy, we just had to plug the two answers in.

For the area of the two triangles, we got 634.2 ft

Our next step, was to find the area of the pizza slices. To do this we had to use SOH CAH TOA. We needed to use this in order to find the angle measurements. We already knew that one of the angle measurements was going to be 45 degrees because one straight line is 180 degrees. We solved for the angle in the middle using SINE, inverse trig. In the red is how I solved for the angles. To find the third angle, we solved for x. We wanted to find a portion of the circle, so we but the angle measurement, (49.53) over 360. This gave us the answer to the pizza slices which was, 3,501.07.

Finally, to get the area of what the cow can graze, we added up all of our areas, and subtracted half of the barn which is 50ft. The answer was

**31,148.5 ft.**

**Evaluation/ Reflection:**

What pushed my thinking was my initial drawing and understanding of the cow problem. I didn't understand how the cow was going to go around the barn. I was just thinking about it going around the barn and it was all very confusing to me until we learned how to maximize the area that the cow was going to walk around. It also helped me a lot the final drawing with the "weird dimple thing" which was the turning point. I think that what I got the most out of this problem was slowing down, and again, not being scared to ask questions, even if the problem was already explained a thousand times, if I had questions, I know I needed to ask them. The group quiz affected me in a positive way. Once we were working on it, I realized how long the process was, and how much of the problem we were still missing. This helped me reach out to my friends after for help.

If I were to grade myself on this problem, I would give myself a solid B. I would give myself this because honestly, there were times when I wouldn't really try if I was confused because I was embarrassed. I would give myself this grade because even when I was confused I would still try to keep up and take as many notes to help me.

If I were to grade myself on this problem, I would give myself a solid B. I would give myself this because honestly, there were times when I wouldn't really try if I was confused because I was embarrassed. I would give myself this grade because even when I was confused I would still try to keep up and take as many notes to help me.