Out of 1200 people polled, $30\%$ do not like radio, and $10\%$ of the people who do not like radio also do not like music.  How many people polled do not like both radio and music?
The number of people who dislike radio is $.3(1200)=360$.  Out of these, the number who also dislike music is $.1(360)=36$ people. So, $\boxed{36}$ people do not like both radio and music.