In a certain region, the electric field is directed outward and is...

Given that the electric field is directed outward and is proportional to the distance $r$ from the origin, we can write:

$\vec{E} = k r \hat{r}$

where $k$ is a positive constant.

The relation between electric field and electric potential is given by:

$dV = -\vec{E} \cdot d\vec{r}$

Substituting the value of $\vec{E}$:

$dV = -(k r \hat{r}) \cdot (dr \hat{r}) = -k r dr$

Integrating both sides from the origin to a distance $r$:

$\int_{0}^{V} dV = -\int_{0}^{r} k r dr$

$V - 0 = -\dfrac{k r^2}{2}$

$V = -\dfrac{k}{2} r^2$

Thus, the electric potential is proportional to $r^2$.