A farmer ties a cow to the corner of a 10' by 10' barn with a rope that stretches for 100'. The farmer, being a strangely OCD man, wants to only plant the exact amount of grass that the cow can graze on. Calculate the area in square feet of how much grass the cow can reach.
Diagrams
When beginning this problem, we first had to create a diagram of the area that the cow could graze on. For me, this problem was very difficult to visualize. As you can see in image (image 1), I was looking at the rope as if it were a more rigid object than it really was. I imagined that after approaching a corner of the barn, the rope would just lose 10' of length and form a new circle that was smaller. This thinking was approaching the real shape, but I misunderstood where the new circle would be centered. The new point would be on the next corner, not on the original. (image 2)
The final diagram looked somewhat peach shaped. In my diagram, I filled in different parts of the circle with diagonal lines. They detail the direction in which the cow started walking. Places where they intersect are where the cow can reach from walking either way around the barn
There is no formula to solve for the area of a peach shape, so we had to split the diagram up into manageable chunks that we could actually solve for. Our split up diagram looked like this (image 3) because we all knew that we could solve for portions of a circle and triangles.
We had some information, but looking at the diagram, we knew that it was not yet enough to plug into the formulas. We still needed the angle of the two pizza slices, the height of the half square triangle, and the base and height of the largest triangle (image 4). We did, however, have enough information to find all of these dimensions!
Solving the problem
**see photos for dimensions given and needed
Formulas needed: Area of a triangle - A = b h (1/2) Area of a circle - A = π r^2 Pythagorean theorem - a^2 + b^2 = c^2 SOH CAH TOA
Techniques practiced: Pythagorean theorem - Use a^2 + b^2 = c ^2 with information given to solve for a missing side length of a right triangle. A&b are the lengths of non hypotenuse sides and c is the length of the hypotenuse. SOH CAH TOA - Use sine, cosine, and tangent to find missing angles or side lengths of a right triangle. Sine = Opposite/Hypotenuse. Cosine = Adjacent/Hypotenuse. Tangent = Opposite/Adjacent. Opposite and Adjacent are terms relevant to the angle you are solving for or have been given. This technique works with the relationship between side lengths and angles in a right triangle. Pizza Slices - In order to solve for the area of a pizza slice, first solve for the area of the whole pizza, then multiply it to the angle of the slice/360. Area of a circle and triangle - See formulas
Lengths given: Sides of the barn (10'x10') 100' of rope 90' length of rope after turning at the corner (this is also the side lengths of the triangle)
The Triangles (image 4)
Use the Pythagorean theorem to find the length of the diagonal cut through the barn (image: solving for a) The answer is 14.14
Split this in half so that we can use the Pythagorean theorem (which only works on right triangles) to find more dimensions in the future. (image: solving for a1 and a2) The answer is 7.07
Use the base length that we calculated and use the Pythagorean theorem to solve for the height of the triangle. In terms of the formula, for this we are plugging in the half base length of 7.07 (a) and the side length of 90 (c) to find the height (b). (image: solving for b) height = 89.7
Knowing that the barn is a square and our base is a diagonal cut across that square, we know that the height of our white triangle will be 7.07. It is half of the line that would cross perpendicular to our base to create the same triangle we are dealing with. So in the image, c can be determined without math.
Solve for the area of the largest triangle using A = b h 1/2 (image: blue + white) A = 635.5
Solve for the area of the barn section of this triangle using the same formula (image: just white) A = 50
Subtract the barn section to only get the area that the cow can actually graze on. (image: (blue + white) - white)
A = 585.5
**the a b and c used when talking about the Pythagorean theorem are different from the variables used in the images
The main circle
Solve for the are of a triangle with a 100' radius. A = 31,415.9
Multiply this number by 3/4
A = 23,561.9
The pizza slices (image 5)
Use cosine (adjacent/hypotenuse, 7.07/90) to find the angle of the large triangle (image: shaded black angle) ϴ = 85.5 degrees
Add 45 degrees because our other angle is half of the 90 degree angle of the square (image: arrow 45) ϴ + 45 = 135.5
Subtract this number from 180 to find the degrees of the pizza slices. (image: blue angle) The number is 49.5. Multiply that number by 2 (2 pizza slices) and set it up as a fraction over 360 degrees
Solve for the area of a circle with a radius of 90'. The answer is 25,446.9
Multiply this to your fraction
A = 6997.9
ADD UP YOUR AREAS! (image 6) A = 31,145.3
Evaluation
I enjoyed the cow problem very much. I liked that I got to be creative and feel like I was solving a puzzle. Once we split up the diagram into pieces and I saw what I had to work with, getting through this problem was a breeze. I got a lot out of the practice that we did on the different trigonometry formulas. I definitely had to work hard to practice my SOH CAH TOA skills. I had never been introduced to this math technique so it was a little bit confusing at first. Eventually I got it down and I now understand how to use it, but I still would like to understand what the calculator does with the information I give it. I enjoyed the group quiz very much because I felt like it allowed me to solidify what we had done in class by taking my group through it step by step. If I hadn't gone over all of that with my group, I don't think I would have gotten 100% on my individual quiz. Because we talked things through, I got through the first page of trig problems very quickly and then sped through the honors portion which was just a much easier version of what we had already done together. I wish that we had done group work of that quality earlier, but to be honest there was a little bit of a language barrier at my table.
If I were to grade myself on this unit, I would give myself an A+. I'm not just saying that because I want a good grade, but I actually think I worked hard on my packets and ended up understanding even variations of the cow problem. I showed on my final exam that I understood the material and even turned in my test before many of my peers. In normal class, I have turned in every math assignment this year on time and have challenged myself by taking honors. However, I think the reason I deserve an A most is for my curiosity. I think outside of the box when I solve problems so that I can experiment with the interaction between the different parts of an equation and I ask genuine questions at the very least 2 or 3 times a day because I really want to understand this material on a deeper level. Math would be very boring without understanding what it was that you were doing. In the cow problem, I believe I figured that out.