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

Question:
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?

Replies:
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:

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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