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Math Help - Calc. app. Q

  1. #1
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    Calc. app. Q

    This is a fairly challenging problem from James multi-variable calc book 5 ed. Here it is: A cow is tied to a silo with radius r by a rope just long enough to reach the other side. What is the total grazing area?

    I've spent quite a good amount of time working on it and come up with some good ideas, but I've hit a bit of a stump, and I'm not getting past it.
    I'm sorry I don't have a scanner, this would be of some use for the diagrams but hopefully this will be enough. I divided the area in two parts (and subdivided those) cutting through the silo. One of those parts of the top half, A1 say, is just a quarter circle of radius pi(r). The other half on the other hand looks a bit like a quarter of a cardioid. This will be called A2. A3 is the area of the top half of the silo. 2(A1+A2-A3)=A

    Diagram.
    http://i1192.photobucket.com/albums/...eulcer/cow.gif

    So now I need a model for A2. I started with the semi-circle here and 0<t<pi x=r(1-cost) and y=rsint. I also figured the length of the rope "against" the silo would be (pi)r-tr. Last of all the tangents of the points on the circle should tell me which direction the rope will be and as I know what the distance left is I figured I could extract the coordinates. (y'=cost/sint). And this is where I'm stuck. I tried using the distance formula here (which was a pain) to find an equation for the coordinates and the x could not be isolated. After coming this far, I'd like to finish, but I'm afraid I may have not picked the best way of representing this. Any insight appreciated.
    Last edited by eulcer; April 9th 2011 at 09:10 AM.
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  2. #2
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    What's the length of the rope used to tie the cow? I'm guessing you're looking for the area bounded by two circles. Polars is therefore the best approach.
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  3. #3
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    Quote Originally Posted by ojones View Post
    What's the length of the rope used to tie the cow? I'm guessing you're looking for the area bounded by two circles. Polars is therefore the best approach.
    Sorry about that. The cow can just reach the other side of the silo (pi(r)). I'll edit above.
    This question happened to come before polar coordinates, so I want to stick with what I have for now, unless you can show me polar.
    Last edited by eulcer; April 9th 2011 at 09:06 AM.
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  4. #4
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    If this question comes before polars then they aren't to be used. You'll have to integrate using rectangular coordinates. I'm wrong anyway about the region being bound by two circles since the rope will start to wrap around the silo. The first and most important step is to identify what the region looks like.
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  5. #5
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    Quote Originally Posted by ojones View Post
    If this question comes before polars then they aren't to be used. You'll have to integrate using rectangular coordinates. I'm wrong anyway about the region being bound by two circles since the rope will start to wrap around the silo. The first and most important step is to identify what the region looks like.
    http://i1192.photobucket.com/albums/...eulcer/cow.gif
    Better?
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  6. #6
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    Yes, but you need to find the equation of this involute. I found your problem in Stewart's book. Right before it is another problem concerning this involute. Solve this one first because it helps with the current question.

    By the way, your comment about polars is incorrect. Polar coordinates are covered in precalculus, not calculus I. You should already be familiar with them! If not, review them because they greatly simplify this calculation.
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  7. #7
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    Quote Originally Posted by ojones View Post
    Yes, but you need to find the equation of this involute. I found your problem in Stewart's book. Right before it is another problem concerning this involute. Solve this one first because it helps with the current question.

    By the way, your comment about polars is incorrect. Polar coordinates are covered in precalculus, not calculus I. You should already be familiar with them! If not, review them because they greatly simplify this calculation.
    I understand I need to find the equation. That's what this was all about. I've come so far into finding one, and I don't know how to put it all together. I should have enough to figure out the equation. I've got the length of the rope around the silo at any point and I know the tangent of the rope to the silo. I will continue with what I've got, if you are not willing to look at this problem. I don't want a given equation, that's the challenge of this problem.

    Polar equations, I'm sorry. You understand what I mean. Parametric equations come before polar equations in the book, no?
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  8. #8
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    As I said, there's question in Stewart that preceeds this one giving you help on how to find the equation of the involute. Do the question; it comes with hints. Also, Stewart is using polar parametric equations to solve the involute problem and so obviously this is something he expects you to know.
    Last edited by ojones; April 9th 2011 at 02:05 PM.
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