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# Speed, Position and Arc Length

Two identical bugs start moving at the same time on a
flat table, each at the same constant speed of 20
cm/min. Assume that initially (i.e. at time t = 0)
bug 1 is located at point (1, 1) and bug 2 is located
at the point (-1, 1). Assume that units in the
xy-plane are measured in meters and time is measured
in minutes. Further assume that the paths of bug 1
and 2 are given respectively by
C1 : x = a*e-alpha*cos (alpha), y = a*e-alpha*sin
(alpha), pi/4 <= alpha < infinity
C2 : x = a*e-beta*cos (beta), y = a*e-beta*sin
(beta), 3(pi)/4 <= beta < infinity
where a and b are constants.

1. Find the exact values of the constants a and b.
2. Find the arc-length of C1 and C2. Use this
information to show that both bugs reach the origin at
the same time To and find the exact value of To.
3. Find the relationship between the parameter alpha
and time t. What is the relationship between beta and
time t?
4. Find the exact distance between bug 1 and bug 2 at
any time t with 0 <= t < To. Use this information to
conclude that bug 1 never captures bug 2 before t =
To.
5. Find the time at which bug 1 is 2 cm from bug 2.
6. How many times does bug 1 wind around the origin
during the time interval 0 <= t <= 0.9999To? Discuss
the motion of bug 1 on the time interval 0.9999To < t
<= To.