The x - t graph of a particle is displayed in the figure. Determine...

The average velocity of a particle is given by the total displacement divided by the total time taken.

$v_{avg} = \dfrac{x(t) - x(0)}{t - 0}$

For the average velocity to be zero, the displacement must be zero, which implies $x(t) = x(0)$.

From the given $x$-$t$ graph, the initial position of the particle at $t = 0$ is $x(0) = 20$ m.

The particle must return to the position $x = 20$ m for the displacement to be zero.

Observing the graph, the horizontal line at $x = 20$ m intersects the curve again at $t = 12$ s.

Therefore, the time $t$ such that the average velocity is zero is $12$ s.