Four-Bug Problem Four bugs sit at the comers of a square carpet L in. on a side. Simultane...

Four-Bug Problem Four bugs sit at the comers of a square carpet L in. on a side. Simultaneously, each starts walking at the same rate of 1 in/sec toward the bug on its right. See Fig. 9(a).

(a) Show that the bugs collide at the center of the carpet in exactly L sec.Hint:Each bug always moves in a direction perpendicular to the line of sight of the bug behind it, so the distance between two successive bugs always decreases at 1 in/sec. The bugs always form a square that is shrinking and rotating clockwise.5

(b) Using the result from (a), but using no calculus, tell how far each bug will travel.

Figure 9 Four-bug problem

(c) Use differential equations to find the paths of the bugs. Simplify the setup by starting the bugs at the four points (±1, 0) and (0, ±1), making  Use Fig. 9(b) to deduce the relationship dr ≈ − r dθ, for sufficiently small dθ.