Cow and fence problem ```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: ``` 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. rcwinther Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs