Interesting word problem posed

• twinhairdryer
In summary: Just rotate the picture 90 degrees.So in summary, the student asked if the area of a cow grazing in a square barn could be computed using a simple area problem. The answer is that the cow can graze in one-fourth of a circle of radius 100, then one-third of a circle of radius 90, then the area between the lines formed by the sides of the barns extended. However, if the cow goes around the barn the other way, she will pick up a little bit more grass symmetric to that last part.
twinhairdryer
So I had a student today ask me if I could come up with a simple 2 dimensional area problem. I was like -sure what is it. The question was:

If a cow were strapped to a tether outside a 10' by 10' square barn, at the corner of a small outlet door, and had a 100' leash/tether, what would the area be that he could cover to graze?

So the obvious is a circular sweep CCW with a 100 square foot keep out area in one direction, but then other? any ideas how to compute this one? thanks newbie

It not a 'simple' problem. It's a complicated problem. Not theory-complicated, just technical-complicated. You'll have to figure out the cow's path with the tether stretched to it's max CCW and CW. At some point they'll probably intersect. Then add up all the area of all the arc sectors making up the path up to that point of intersection.

It's best to start by drawing a picture. Obviously the cow can graze in three-fourths of a circle of radius 100, the three quadrants not containing the barn. Now stretch the tether across the front of the barn. You can now think of the cow as tethered at the other corner and the tether is now 100-10= 90 feet long. The cow can graze in one-fourth of a circle of radius 90. Now move to the third corner. The tether is now 90- 10= 80 feet long but you've already included a portion of this quadrant in your first three-quarters of a circle. The "new" grass is the portion of a circle of radius 80 "behind" the barn: between the lines formed by the sides of the barns extended. That is equal to the area bounded by the lines x= 0, x= 10, y= 0 and the curve x2+ y2= 802.

Added in edit: oops. Just as Dick said, if the cow goes around the barn the other way, she will pick up a little bit more grass symmetric to that last part. Because it is symmetric do this: find the area bounded by the lines x= 0, x= 5, y= 0 and x2+ y2= 80[/sup]2[/sup] and double it to get both parts.

Wrong again! It is not symmetric. Going around the barn the other way, the cow only has to pass the side of the barn, not front and side. The point where the two circles meet in back will be the intersection of x2+ y2= 802 and (x-10)2+ y2= 902. And you will need to find the area bounded by x= 0, x= a, where a is the x coord on that intersection, y= 0 and x2+ y2= 802, find the area bounded by x= a, x= 10, y= 0 and (x- 10)2+ y2= 902, and add.

Last edited by a moderator:
Actually, there is a usable symmetry. The path is symmetric across the diagonal connecting the corner of the barn the cow is tethered to and the opposite corner. That helps.

1. What is the most important step in solving a word problem?

The most important step in solving a word problem is to carefully read and understand the problem. It is important to identify the key information and what is being asked in order to come up with an appropriate solution.

2. How do I know which operation to use in a word problem?

To determine which operation to use in a word problem, you must pay attention to what the problem is asking for. If the problem is asking for a total, you may need to add. If it is asking for a difference, you may need to subtract. If it is asking for a product, you may need to multiply. And if it is asking for a quotient, you may need to divide.

3. Can I use a calculator to solve word problems?

Yes, you can use a calculator to solve word problems, but it is important to use it as a tool and not solely rely on it. It is important to understand the concepts and steps involved in solving the problem, rather than just plugging in numbers into a calculator.

4. How can I check if my answer to a word problem is correct?

You can check if your answer to a word problem is correct by re-reading the problem and plugging your answer back into the problem to see if it makes sense. You can also try solving the problem using a different method to see if you get the same answer. Additionally, you can ask someone else to solve the problem and compare your answers.

5. Is it possible to solve a word problem without using numbers?

Yes, it is possible to solve a word problem without using numbers. This is often referred to as "solving symbolically" and involves using variables to represent unknown quantities and writing equations to represent relationships between these quantities. However, in most cases, solving symbolically will eventually require the use of numbers to find a specific solution.

Similar threads

• Precalculus Mathematics Homework Help
Replies
5
Views
4K
• Introductory Physics Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• General Discussion
Replies
13
Views
3K
• General Math
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
9
Views
2K
• General Engineering
Replies
2
Views
2K
• General Math
Replies
17
Views
2K