Problem Statement:
At the very beginning of the year we were presented with a word problem, this is it:
There is a square barn that is 10ft by 10ft. A cow is attached to one of the barns corners with a rope that is 100 ft long. What is the area the cow can graze?
There is a square barn that is 10ft by 10ft. A cow is attached to one of the barns corners with a rope that is 100 ft long. What is the area the cow can graze?
Process:
First attempt:
When we were first given this problem we were told to just try to solve it without any information from the teacher. I knew that the cow was going to be able to walk at least ¾ of a circle around the barn, so the first thing I drew was a square with a rope drawn onto the corner, then I drew how I thought the rope would move when the cow walked. I ended up getting a shape very close to what we would be actually working with.
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Final, accurate diagram:
In my final diagram you see the 10ft by 10ft barn with 100 feet of rope tied to one corner. Then I drew the ¾ of the circle until it wouldn't be able to go around the barn without having to fold. Once you got to the point where the rope folds and the cow only has 90 ft left (because the 10ft barn stops the cow from being able to use the entire 100ft to move to areas he has not grazed yet) then the dimple in the circle forms. After this I drew a Triangle with the tip of the dimple and, for the sake of having right angles, had the base be at half of the barn (we subtracted the area of the barn when we found the final triangle area)
Splitting the odd shape into 3/4 of a circle, a triangle minus part of the barn, and two rounded triangles, made it easier to find the total are since the shapes were already familiar and we knew how to find the area of. To be able to solve the problem you need to know the dimensions of the barn how long the rope is, area finding formulas, and other math skills i'll explain in the solution. |
Solution:
Total area:
To find the total area the cow could graze you have to find the area of all the individual shapes we created and then add them all together
Formulas you'll need to know: Area of a circle: πR^2 Area of Triangle: Base*Highet divided by 2 Pythagreon therom: a^2+b^2=c^2 SOH CAH TOA: Sine Opp/Hyp, Cosine Adj/Hyp, Tan Opp/Adj |
Area of 3/4 of circle:
What I did to find the are of the 3/4 of the circle was to first use the formula foe finding the are of a complete circle : πR^2
I then plugged in 100 for the radius (because from the middle point of the circle and the edge the length is 100ft) You have to remember to square (multiply the number by the same number) the number first before you multiply it by π because of order of opperations. After you get the area of the whole circle you multiply the total by 0.75 because that is the percentage of the circle that we actually have. The area of that part of the shape ends up being 23,562. |
Area of the inner triangle:
Before you can find the area of the triangle you have to find the base of the triangle and the height of the triangle.
Base: To find the base I used the Pythagorean theorem (a^2+b^2=c^2) In this case a and b were both 10 Because of the piece of the barn. Within the triangle c was the missing side. C is always going to be the hypotenuse, which is the longest side of a triangle. Once you solve 10^2+10^2=C^2, you end up with C^2 = 200 To find what c is (the base) you would have to square root 200. You end having the base be 14.14. Since we have this triangle split in half, each side would be 7.07. Height: To find the height of the triangle you have to use the Pythagorean theorem yet again. In this case A is what you are trying to find. Since it is caught in half b is 7.07, c is 90 because it is the longest side. It is 90 this time because the longest side is equal to the side of the rope. After you solve A^2+7.07=90^2 you have A^2=8,050. To find the height you find the square root of 8,050. The height of the triangle is 89.7 To find the final area of the triangle, you just use do the base * Height divided by to. Here we will do 14.14 * 89.7. You then divide the answer by 2 to get the area of the total triangle. However since half of the barn is inside of the triangle we just found the are of, we have to subtract the area of the barn form the total triangle area. Since only half of the barn needs to be subtracted, you take away 50. The area of the triangle we needed is : 584.2 |
Area of the two rounded triangles:
Self evaluation: |
Before you can find the area of this final shapes are you have to find the angles. To be able to find the are you need to know which function to use. In this case you would use sine, because you have the opposite and hypotenuse. To find an angle you have to do the inverse function, by pressing the 2nd button on the calculator. After you find one of the missing angles, you add 45 to it. ( you know you have a 45 degree angle because the triangles angle was half of the barns angle, which was 90)
Then you have to subtract the sum from 180 (because all angels have to add up to 180) to find the missing angle. The final l angle is 49/7. The next step is to find the are of these pizza slice looking shapes. since this shape is like part of the circle you would find the area of the circus using π90^2 (90 would be the radius because in this case you have 10 ft. less rope than in the 3/4 circle because the barn folds it.) Once you find the are of a circle, you multiply it by 0.138. You get this number by dividing the angle 49.7 by 360, this helps us find what percentage of the circle we have. When you find the are of one of the shapes all you have to do is add them together. 3,512 + 3,512 = 7,024 this is the final area of the shape. The final thing left to do is add all the areas together. I ended up with a total area of 31,170 |
Words problems like this are always the most challenging for me. It was a struggle to understand at first but with the help of the work sheets, Mr. B and my classmates I ended up understanding the problem. I think I got a lot out of this problem because I feel like I worked very confidently in this problem and did not doubt my mathematical ability. This made it possible for me to dive in deeper and have a stronger understanding of the topics needed to solve the problem. The group quiz followed by an individual test really worked for me I would say. We were able to work as a team and try to get the answer. Although we did not get the right answer, there was never a moment where we weren't working together and asking questions. The individual test was a good addition to it because the group would show our group work skills, which are very important and the individual would showcase what each person understood. I would personally give myself an A because I was consistently doing my working and trying to understand. I asked for help when needed but I did not give up ever, which used to be something I would do a lot. I think my "growth mindset" really shone through while doing this problem.