Holy Trashcan
12-06-2007, 07:12 PM
i came up with a pretty basic geometry/calc problem that i've tried solving, but either ran into computational hemorrhages or simply got stuck. i'd post what i've already done but i've tried many different approaches, and i think it's probably best to approach it with a clean slate.
Consider the square WXYZ with side length 1. Point A lies on side WX, point B lies on side XY, and the length of AB = 1. Point P lies on AB such that the line ZP is perpendicular to AB. Let θ be the angle between ZP and ZY. Compute the length of XP in terms of θ.
As θ ranges from 0 to π/2, the locus of points P divides the square into two sections. This problem comes from trying to find the area of the smaller of these two sections, which just involves a simple integration. Of course, if anyone has a more elegant way of finding this area, feel free.
Consider the square WXYZ with side length 1. Point A lies on side WX, point B lies on side XY, and the length of AB = 1. Point P lies on AB such that the line ZP is perpendicular to AB. Let θ be the angle between ZP and ZY. Compute the length of XP in terms of θ.
As θ ranges from 0 to π/2, the locus of points P divides the square into two sections. This problem comes from trying to find the area of the smaller of these two sections, which just involves a simple integration. Of course, if anyone has a more elegant way of finding this area, feel free.