On January 22,2003 the distance between Sun and Pioneer 10 was 21,275,527,680,000m=2.12*10^13m
now on january 2009 the pioneer 10 has alreay covered 6 yrs of time
so , 6yrs= 1.892*10^8sec
the relative velocity V=12.24Km/s and t=1.892*10^8sec
then in 6 years the pioneer 10 has covered distance d=12.24*1.892*10^8Km=23.1508080*10^8km=231508080000m
so overall distance=21275527680000m + 2315080800000m=.2150*10^14m
Aldebaran was in the distance 6.15*10^7m
then in 6 years=1.892*10^8sec the distance will be 6.15*10^7*1.892*10^8m=1.16358*10^16m
in 2009 the distance between aldebaran and pioneer 10 is=1.16358*10^16-.2150*10^14m=1.1614*10^16m
the speed of pioneer= 12.24km/s=12240m/s
so pioneer covers 12240m in one sec
pioneer covers 1m in 1/12240sec
then pioneer covers 1.1614*10^16m in 1.1614*10^16*1/12240sec=.0000947*10^16sec=3.029*10^-12yr
For your answers, all of your steps seem correct!
Does your teacher check for the exact answer? If you are a hundredth of a thousandth off, does your teacher consider it wrong?
Your answers and solution steps seem accurate, so you should be fine.
You can see what other people say though.
Here is how long it should take Pioneer 10 spacesraft to reach the star Aldebaran. According to Wikipedia, the distance to Aldebaran is ~65.3 light years from the Sun.
The speed of light is: 299,792.458 km/second.
The speed of Pioneer 10 is 12.24 km/second.
299,792.458 / 12.24 =24,492.85 times is the speed of light faster than the speed of the spacecraft.
Therefore, it will take Pioneer 10:
24,492.85 x 65.3 =1,599,383 Earth years to reach Aldebaran. Make sure that your final answer agrees roughly with this number.