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Name: david j downing
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

If a cow is tied to a fence post that is part of a circular field, how long would the rope have to be in relation to the radius of the circle so that the cow could eat half of the grass in the field. (Assuming that there is grass in the field) I believe this one needs numerical analysis. Am I correct?

You did not say where the fence post is, but I assume it is on the edge of the circular field. I do not think that this problem can be done without using an integral; after setting up the problem and doing the integration, I end up with the following equation to solve:
    asin[k*sqrt(4 - k^2)/2] + k^2*asin[sqrt(4 - k^2)/2]
                            - k*sqrt(4 - k^2)/2 - pi/2 = 0

which can be simplified somewhat to
   acos(1 - k^2/2) + k^2*acos(k/2) - k*sqrt(4 - k^2)/2 - pi/2 = 0

where k is the ratio of the rope length to the radius of the circular field, k^2 means "k squared", "asin" is the inverse sine function, and "acos" is the inverse cosine function. And yes, a numerical method is needed in order to find an approximate value of k for either form of the k-equation (that is, it cannot be solved in closed form). There are a number of methods available; I used Newton's method and obtained a value of 1.1587, correct to four decimals.


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