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 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:
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.
Maximum Rectangle
Problem Statement:
A rectangle has one corner on the graph of y=16x^2 , another at the origin, a third on the positive yaxis, and the fourth on the positive xaxis. If the area of the rectangle is a function of x, what value of x yields the largest area of the rectangle?
Process:

The first thing we did in order to fully understand the problem, we started by writing down a few questions. We shared them outloud and most of them were answered. After this, we plugged in numbers in our given equation which was y = 16  x^2. We would plug in the the numbers in the x intercepts for x. This would give us points that would soon form into a parabola when we connected them. We knew that the y intercept was going to be 16. Inside the parabola we had to find a few rectangles, and find which one had the largest area. The rectangle with the largest area was the 2 by 12 which gave us an area of 24. This was our ESTIMATED area, not the final one. We noticed that we had to go by really small decimals, and we didn't want to keep going on forever, so we had to come up with an equation. The equation was A= 16x  x^3. It was now cubed because we needed to factor it out. Then, we needed to find an estimate for the perimeter, and again, like the last problem we needed to form an equation for it. We found the equation to be P=2x^2 +2x + 32. We know this because in order to find the perimeter, you need to add all the sides. Our first equation was P= 2( x+y), but then we plugged in our givens which were x and y. We knew that these equations were correct because once we graphed them, we were able to get both the maximum perimeter and maximum area.

Solution:
The solution for the maximum perimeter was 32.5 with the side lengths of (0.5, 15.75). We found this by using the equation we had, and turned it into vertex form. Once we found what the vertex was, we found out that the x was the width and the y was the maximum perimeter.
The solution for the maximum area was between 21 and 24. We used our equation to find it and we got 24.633609. Using this, our side lengths were 2.31 and 10.6639. The way we got this answer was by rounding to the hundredths place.
The solution for the maximum area was between 21 and 24. We used our equation to find it and we got 24.633609. Using this, our side lengths were 2.31 and 10.6639. The way we got this answer was by rounding to the hundredths place.
Group Test/ Individual Test:
Our group prepared for this group quiz by going over the Maximum Rectangle problem, then we worked on the practiced doing the practice quiz. The practice quiz was the same thing just with one different number. I honestly do not think we understood the whole problem very well. We were overthinking the whole thing a lot and we were making tables that were unnecessary. We struggled a lot with finding the actual area, we understood how to find the equations but not the final answer. My understanding for the individual quiz was very good, I was actually surprised. I think I was able to understand it because I didn't have my peers to make me overthink everything. The overall experience of the group quiz was good, but I think it distracted me a lot.
Evaluation/ Reflection:
What really pushed my thinking was how much my group struggled to fully understand this problem. I don't think I've ever been in a group where all of us are confused. This pushed my thinking because it made me put more effort in understanding the problem and trying to help my partners. I think that I got the most out of being able to solve this problem in multiple ways. For example, "plug and chug", making a table, graphing it, and actually solving the equation. I think I would give myself and A because I really did try my hardest to try to understand this problem even though I was confused most of the time.