SOLUTIONS TO EXERCISE 18

Since Venus, Earth and the Sun move relative to each other, Venus can
also have phases as seen from Earth.  Refer to diagrams at the URL
http://astrosun.tn.cornell.edu/courses/astro201/venus_phase.htm for details.

One might expect that, since Venus takes 243 days to go around the Sun once,
that a quarter phase of Venus would take one quarter of that time, that is
about 61 days.  However, since Earth takes 365 days to go around the Sun once,
and it's moving in the same direction as Venus, both the orbits of Earth and 
Venus must be taken into account to compute the duration of phases.  The 
"synodic period" is the amount of time a full cycle of phases will occur;
this is about 584 days.  (See, for example, the URLs
http://www.mira.org/fts0/planets/093/text/txt001x.htm or
http://www.ssec.wisc.edu/outreach/SPARK/venus.html for more details.)
So the time we see Venus go from new to first quarter is 584/4 = 146 days.