Extra-Credit. (2 pts.) Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at al times They approach the center of the square along spiral paths. (Show each step by hand.) (a) Find the polar equation of a bug's path assuming the pole (i.e. the origin) is at the center of the square Cl Hint: Use the fact that the line joining one bug to the next is tangent to the bug's path. This means that at all times, the tangent lines create a square. Furthermore, and this is im portant, a square looks the same after a Ï/2 counter-clockwise a spin. So number the "Bugs" 1 through 4 starting with the bug that begins in the top right-hand corner and counting counter clockwise Suppose that Bug # 1 follows a path given by r = f(0) for θ Ï/4. Given any value for θ, Bug # 1 will be at the point (x1 , y1) = (f(0) cos θ,f(θ) sin θ) Of course, Bug # 2 will then be at (z2.Y) = (f (0) cos(θ + Ï/2), f(0) sin(θ + Ï/2)) The tangent line at Bug # 1 goes through Bug # 2. So ⦠(b) Find the distance traveled by a bug by the time it reaches the other bugs at the center
