
Special Relativity
In the distant future a spacecraft, travelling at 0.1c, is carrying a group of people from Earth to Saturn. Half way through the voyage a terrorist threat is received indicating that there is a bomb onboard set to go off in 6 hours (ship time) after the signal is received if certain demands are not met. Planning to gain more time to search the ship and find, and jettison, the bomb the captain immediately accelerates the ship upto 0.69c. Assuming that the threat is real how long do the crew now have to find and jettison the bomb?
[Specify units of 'hours', 'mins' or 's' (without quotes) in your answer.]
What I did was
found beta = (vfvi)/c (0.69c0.1c) / 3 x10^8
that gave me a beta value of 0.590
then I calculated gama
1/ (1beta)^(1/2) therefore 1/(10.590)^(1/2)
to give me 1.56
then I said time*(gama  1)
I put it in seconds and mins and neither worked
6hrs * 3600 sec/hr * 9 (1.561)
to give me 12096 sec or 201.6 mins
and this is not correct, where have I gone wrong?

I believe you should be using
$\displaystyle t = \frac{t_0}{\sqrt{1  \beta^2}}$
$\displaystyle t = \frac{6\text{hrs}}{\sqrt{1  0.59^2}} = 7.4 \text{hrs}$