Problem Statement
What is the cow problem? The problem we had to solve was that a farmer had a 10ft x 10ft barn and a cow which was attached by a rope to one corner of the barn that measured 100ft and he wanted to find out how much area the cow could graze. To get a better visual you can see the image of the problem to the right, it was a unique shape which is what made finding the area more difficult. You can see the process we went through to find a solution below. |
Process
First attemptsWhen we were first introduced to the problem Mr. Carter told us to sketch out ideas that could possibly help us find the area, you can see a few of my ideas to the left. The one with all the cubes was me trying to use something that we already knew how to find the area of which was a square and see how many could fit inside the shape but we ran into a problem which was that we would eventually end up with all these weird shaped squares which we didn't know how to find the area of. The diagram to the left of the one with all the squares was me attempting to break down the diagram into triangles which we also knew how to find the area of but we would run into the same problem and end up with shapes we couldn't find the area of. So finally as a class we started to break down the diagram and came up with the two designs on the bottom which had shapes we could find the area of.
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Breaking down the diagram
Below you can see the step by step process we went through to try and break down the circle into manageable pieces.(Going form left to right) The first diagram was broken down by the two lines with red on them and that would leave us with 3/4 of a circle. For the second diagram we added the two lines with red lines on them and that helped us because we knew the length of the lines (90ft) and in the next diagram you can see we were able to create a triangle by making a line right between the barn. I explained how we made a triangle with diagram two and three but we cant get the area of a triangle without a height so we added the final line which represented the height. Breaking down the diagram into these sections was really helpful because it left us with pieces that we could find the area of such as 3/4 of a circles, other parts of circles and triangles.
Final diagram / dimensions
This is the final diagram we came up with and it includes all the measurements to find the area of each section. Some formulas we used to find some of the lengths to sides and angles include a²+b²=c² and some trigonometric functions. We used the pythagorean theorem(a²+b²=c²) to find the missing side length in blue and that lead us to find the side length in red. After finding the side lengths for all parts of the diagram it allowed us to find missing angles using sin( , cos(, and tan(. Once we had all the angles and lengths we were able to start finding the area of all the pieces of the diagram and get to a solution which you can see below. |
Solution
Total Area: 31,132.9ft²
The final answer we got was 31,132.9ft² and we got by adding up all the areas of each section which you can see below and the work I showed above also played a big part in finding the area because without the lengths and angles we couldn't of used the formulas in the first place
Area of each sectionPink: 23,550ft² - We got this area by using A=π x r² ( 3.14 x 100² = 31,400) and since we knew that it was only 3/4 of a circle after we multiplied 31,400 x 3/4 and got 23,550.
Yellow: 3,499.3ft² - We found the area of this section using the same A=π x r² but here the numbers changed a little. 3.14 x 90² = 25,434 after we got this answer we multiplied it by 49.53/360 since it was 49.53/360 of the circle. Blue: 317.16ft² - To find the area of the blue triangle we used the formula A= 1/2 (bxh) which will give you the area of a triangle, the numbers we used were 1/2 (5√2 x 87.72) = 317.16ft² Orange: 50ft² Since orange was a square in the beginning finding the area was easy all we had to do was length x width (10 x 10=100) and cut that area in half which is 50. |
Math needed to solve
Below is the math and some formulas you needed to use in order to solve this problem, we learned most of this through the handouts and it really helped to solve the cow problem.
- a²+b²=c²
-trigonometric functions (sin, cos, tan)
- A=π x r²
-A= 1/2 (bxh)
-Subtraction, addition, multiplication
- a²+b²=c²
-trigonometric functions (sin, cos, tan)
- A=π x r²
-A= 1/2 (bxh)
-Subtraction, addition, multiplication
Evaluation / Reflection
I really enjoyed working on this problem especially in the beginning because I had no idea how we were going to solve it which really pushed my thinking. I remember thinking how to break down the diagram and what math we could use to solve it and I was lost I didn't know where to start but as soon as we broke down the diagram as a class everything started coming together. I think the group quiz didn't have much of an effect on me but if it did it wasn't negative for sure. If I were to grade myself on this I would give myself an A- because I feel really good about understanding and explaining the problem and all the math that is included in the problem but I messed up on one problem in the test so that's why I give myself the A-.