The total duration of the meeting \( t \) consists of the time the president speaks and the time she spends answering questions.
According to the description:
- The president speaks for 20 minutes.
- The time spent answering questions is represented by \( q \).
Therefore, we can express the total time \( t \) as: \[ t = 20 + q \]
If we are given that \( q = 10 \) minutes, we can substitute this value into the equation:
\[ t = 20 + 10 = 30 \]
Thus, the value of \( t \) when \( q = 10 \) is: \[ \boxed{30} \]