Three friends are dividing up a pizza that has not been pre sliced, but only wish to make two straight line cuts, dividing the pie into three even pieces. If the center of the pizza is on the origin of a graph, the radius is six inches, and the pizza is to be divided using only vertical cuts, how far from the origin must the vertical cuts be made?

Let the pizza have radius of 1 unit.  Then putting the center of the pizza at (0,0), integrating sqrt(1-x^2) from x=0 to x=X gives the area of the top right quarter of the piece that lies between x=-X and x=+X. We seek X such that this integral is  then 1/4 of the area of any one of the three even pieces, that is, 1/12 of the pie.  The integral is (arcsin(X) + X*sqrt(1-X^2))/2 and equating this to pi/12 and solving numerically for X gives X=.26493195640619815.

Posted by Richard on 2005-12-30 12:49:07